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Properties of Exponents
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Examples
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1 Simplify the expression:
A
B
C
D
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2 Simplify the expression:
A
B
C
D
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4 Simplify the expression:
A
B
C
D
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6 Simplify the expression:
A
B
C
D
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Writing Numbers in Scientific Notation
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Writing Large Numbers in
Scientific Notation
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Here are some different ways of writing 6,500.
6,500 = 6.5 thousand6.5 thousand = 6.5 x 1,0006.5 x 1,000 = 6.5 x 103
which means that
6,500 = 6.5 x 103
6,500 is the standard form of the number6.5 x 103 is scientific notation
These are two ways of writing the same number.
Scientific Notation
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6.5 x 103 isn't a lot more convenient than 6,500.
But let's do the same thing with 7,400,000,000which is equal to 7.4 billionwhich is 7.4 x 1,000,000,000which is 7.4 x 109
Besides being shorter than 7,400,000,000, its a lot easier to keep track of the zeros in scientific notation.
And we'll see that the math gets a lot easier as well.
Scientific Notation
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Scientific notation expresses numbers as the product of:
a coefficient and 10 raised to some power.
3.78 x 106
The coefficient is always greater than or equal to one, and less than 10.
In this case, the number 3,780,000 is expressed in scientific notation.
Scientific Notation
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Express 870,000 in scientific notation
1. Write the number without the comma.
2. Place the decimal so that the first number will be less than 10 but greater than or equal to 1.
3. Count how many places you had to move the decimal point. This becomes the exponent of 10.
4. Drop the zeros to the right of the right-most non-zero digit.
870000
870000 x 10.
870000 x 10.12345
8.7 x 105
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Express 53,600 in scientific notation
53600
53600 x 10. ?
53600 x 10.1234
5.36 x 104
1. Write the number without the comma.
2. Place the decimal so that the first number will be less than 10 but greater than or equal to 1.
3. Count how many places you had to move the decimal point. This becomes the exponent of 10.
4. Drop the zeros to the right of the right-most non-zero digit.
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284000000
284000000 x 10. ?
.1234
?284000000 x 10
7 6 58
2.84 x 108
Express 284,000,000 in scientific notation?
1. Write the number without the comma.
2. Place the decimal so that the first number will be less than 10 but greater than or equal to 1.
3. Count how many places you had to move the decimal point. This becomes the exponent of 10.
4. Drop the zeros to the right of the right-most non-zero digit.
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7 Which is the correct coefficient of 147,000 when it is written in scientific notation?
A 147
B 14.7C 1.47D .147
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8 Which is the correct coefficient of 23,400,000 when it is written in scientific notation?
A .234
B 2.34C 234.D 23.4
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9 How many places do you need to move the decimal point to change 190,000 to 1.9?
A 3
B 4C 5D 6
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10 How many places do you need to move the decimal point to change 765,200,000,000 to 7.652?
A 11
B 10C 9D 8
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11 Which of the following is 345,000,000 in scientific notation?
A 3.45 x 108
B 3.45 x 106
C 345 x 106
D .345 x 109
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12 Which of these is not a large number in scientific notation?A .34 x 108
B 7.2 x 103
C 8.9 x 104
D 2.2 x 10-1
E 11.4 x 1012
F .41 x 103
G 5.65 x 104
H 10.0 x 103
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Content/writing spaceThe mass of the solar system is 300,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 kg (How do you say that number?)
(That's 3, followed by 53 zeros)
What is this number in scientific notation?
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More Practice
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Express 9,040,000,000 in scientific notation
9040000000
9040000000 x 10. ?
. ?9040000000 x 10
9 8 7 6 5 4 3 2 1
9.04 x 109
1. Write the number without the comma.
2. Place the decimal so that the first number will be less than 10 but greater than or equal to 1.
3. Count how many places you had to move the decimal point. This becomes the exponent of 10.
4. Drop the zeros to the right of the right-most non-zero digit.
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Express 13,030,000 in scientific notation
13030000
13030000 x 10. ?
. ?13030000 x 10 7 6 5 4 3 2 1
1.303 x 107
1. Write the number without the comma.
2. Place the decimal so that the first number will be less than 10 but greater than or equal to 1.
3. Count how many places you had to move the decimal point. This becomes the exponent of 10.
4. Drop the zeros to the right of the right-most non-zero digit.
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Express 1,000,000,000 in scientific notation
1000000000
1000000000 x 10. ?
. ?1000000000 x 10 9 8 7 6 5 4 3 2 1
1 x 109
1. Write the number without the comma.
2. Place the decimal so that the first number will be less than 10 but greater than or equal to 1.
3. Count how many places you had to move the decimal point. This becomes the exponent of 10.
4. Drop the zeros to the right of the right-most non-zero digit.
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13 Which of the following is 12,300,000 in scientific notation?
A .123 x 108
B 1.23 x 105
C 123 x 105
D 1.23 x 107
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Writing Small Numbers in
Scientific Notation
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Express 0.0043 in scientific notation
0043
0043 x 10. ?
0043 x 10.1 2 3
?
4.3 x 10-3
1. Write the number without the decimal point.
2. Place the decimal so that the first number is 1 or more, but less than 10.
3. Count how many places you had to move the decimal point. The negative of this numbers becomes the exponent of 10.
4. Drop the zeros to the right of the right-most non-zero digit.
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Express 0.00000832 in scientific notation
00000832
00000832 x 10. ?
00000832 x 10.1 2 3 4
?
5 6
8.32 x 10-6
1. Write the number without the comma.
2. Place the decimal so that the first number will be less than 10 but greater than or equal to 1.
3. Count how many places you had to move the decimal point. This becomes the exponent of 10.
4. Drop the zeros to the right of the right-most non-zero digit.
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Express 0.0073 in scientific notation
0073 x 10.1 2 3
?
0073
0073 x 10. ?
7.3 x 10-3
1. Write the number without the comma.
2. Place the decimal so that the first number will be less than 10 but greater than or equal to 1.
3. Count how many places you had to move the decimal point. This becomes the exponent of 10.
4. Drop the zeros to the right of the right-most non-zero digit.
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14 Which is the correct decimal placement to convert 0.000832 to scientific notation?
A 832
B 83.2C .832D 8.32
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15 Which is the correct decimal placement to convert 0.000000376 to scientific notation?
A 3.76B 0.376C 376.
D 37.6
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16 How many times do you need to move the decimal point to change 0.00658 to 6.58?
A 2
B 3C 4D 5
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17 How many times do you need to move the decimal point to change 0.000003242 to 3.242?
A 5B 6C 7D 8
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18 Write 0.00278 in scientific notation.
A 27.8 x 10-4
B 2.78 x 103
C 2.78 x 10-3
D 278 x 10-3
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19 Which of these is not a small number in scientific notation?
A .34 x 10-8
B 7.2 x 10-3
C 8.9 x 104
D 2.2 x 10-1
E 11.4 x 10-12
F .41 x 10-3
G 5.65 x 10-4
H 10.0 x 10-3
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More Practice
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Express 0.001002 in scientific notation
001002 x 10.1 2 3
?
001002
001002 x 10. ?
7.3 x 10-3
1. Write the number without the comma.
2. Place the decimal so that the first number will be less than 10 but greater than or equal to 1.
3. Count how many places you had to move the decimal point. This becomes the exponent of 10.
4. Drop the zeros to the right of the right-most non-zero digit.
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Express 0.00092 in scientific notation
00092 x 10.1 2 3 4
?
00092
00092 x 10.?
9.2 x 10-4
1. Write the number without the comma.
2. Place the decimal so that the first number will be less than 10 but greater than or equal to 1.
3. Count how many places you had to move the decimal point. This becomes the exponent of 10.
4. Drop the zeros to the right of the right-most non-zero digit.
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Express 0.0000012 in scientific notation
0000012 x 10.1 2 3 4 5 6
?
0000012
0000012 x 10. ?
1.2 x 10-6
1. Write the number without the comma.
2. Place the decimal so that the first number will be less than 10 but greater than or equal to 1.
3. Count how many places you had to move the decimal point. This becomes the exponent of 10.
4. Drop the zeros to the right of the right-most non-zero digit.
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20 Write 0.000847 in scientific notation.
A 8.47 x 104
B 847 x 10-4
C 8.47 x 10-4
D 84.7 x 10-5
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Converting to
Standard Form
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Express 3.5 x 104 in standard form
35,000
1. Write the coefficient.
2. Add a number of zeros equal to the exponent: to the right for positive exponents and to the left for negative.
3. Move the decimal the number of places indicated by the exponent: to the right for positive exponents and to the left for negative.
4. Drop unnecessary zeros and add comma, as necessary.
3.50000
3.5
35000.0
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Express 1.02 x 106 in standard form
1,020,000
1.02000000
1.02
1020000.00
1. Write the coefficient.
2. Add a number of zeros equal to the exponent: to the right for positive exponents and to the left for negative.
3. Move the decimal the number of places indicated by the exponent: to the right for positive exponents and to the left for negative.
4. Drop unnecessary zeros and add comma, as necessary.
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Express 3.45 x 10-3 in standard form
0003.45
3.45
0.00345
1. Write the coefficient.
2. Add a number of zeros equal to the exponent: to the right for positive exponents and to the left for negative.
3. Move the decimal the number of places indicated by the exponent: to the right for positive exponents and to the left for negative.
4. Drop unnecessary zeros and add comma, as necessary.
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Express 2.95 x 10-5 in standard form
000002.95
2.95
0.00000295
1. Write the coefficient.
2. Add a number of zeros equal to the exponent: to the right for positive exponents and to the left for negative.
3. Move the decimal the number of places indicated by the exponent: to the right for positive exponents and to the left for negative.
4. Drop unnecessary zeros and add comma, as necessary.
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21 How many times do you need to move the decimal and which direction to change 7.41 x 10-6 into standard form?
A 6 to the rightB 6 to the leftC 7 to the rightD 7 to the left
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22 How many times do you need to move the decimal and which direction to change 4.5 x 1010 into standard form?
A 10 to the rightB 10 to the leftC 11 to the rightD 11 to the left
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23 Write 6.46 x 104 in standard form.
A 646,000
B 0.00000646C 64,600D 0.0000646
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24 Write 3.4 x 103 in standard form.
A 3,430
B 343C 34,300
D 0.00343
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25 Write 6.46 x 10-5 in standard form.
A 646,000
B 0.00000646C 0.00646D 0.0000646
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26 Write 1.25 x 10-4 in standard form.
A 125
B 0.000125C 0.00000125D 4.125
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27 Write 4.56 x 10-2 in standard form.
A 456B 4560C 0.00456D 0.0456
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28 Write 1.01 x 109 in standard form.
A 101,000,000,000B 1,010,000,000C 0.00000000101D 0.000000101
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Comparing Numbers Written in
Scientific Notation
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First, compare the exponents.
If the exponents are different, the coefficients don't matter; they have a smaller effect.
Whichever number has the larger exponent is the larger number.
Comparing numbers in scientific notation
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Comparing numbers in scientific notation
< >
9.99 x 103 2.17 x 104
1.02 x 102 8.54 x 10-3
6.83 x 10-9 3.93 x 10-2
=
When the exponents are the different, just compare the exponents.
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If the exponents are the same, compare the coefficients.
The larger the coefficient, the larger the number (if the exponents are the same).
Comparing numbers in scientific notation
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Comparing numbers in scientific notation
5.67 x 103 4.67 x 103
When the exponents are the same, just compare the coefficients.
4.32 x 106 4.67 x 106
2.32 x 1010 3.23 x 1010
< >=
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29 Which is ordered from least to greatest?
A I, II, III, IV
B IV, III, I, II
C I, IV, II, III
D III, I, II, IV
I. 1.0 x 105
II. 7.5 x 106
III. 8.3 x 104
IV. 5.4 x 107
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30 Which is ordered from least to greatest?
A I, II, III, IV
B IV, III, I, II
C I, IV, II, III
D I, II, IV, III
I. 1.0 x 102
II. 7.5 x 106
III. 8.3 x 109
IV. 5.4 x 107
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31 Which is ordered from least to greatest?
A I, II, III, IV
B IV, III, I, II
C III, IV, II, I
D III, IV, I, II
I. 1 x 102
II. 7.5 x 103
III. 8.3 x 10-2
IV. 5.4 x 10-3
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32 Which is ordered from least to greatest?
A II, III, I, IV
B IV, III, I, II
C III, IV, II, I
D III, IV, I, II
I. 1 x 10-2
II. 7.5 x 10-24
III. 8.3 x 10-15
IV. 5.4 x 102
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33 Which is ordered from least to greatest?
A I, II, III, IV
B IV, III, I, II
C I, IV, II, III
D III, IV, I, II
I. 1.0 x 102
II. 7.5 x 102
III. 8.3 x 102
IV. 5.4 x 102
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34 Which is ordered from least to greatest?
A I, II, III, IV
B IV, III, I, II
C I, IV, II, III
D III, IV, I, II
I. 1.0 x 106
II. 7.5 x 106
III. 8.3 x 106
IV. 5.4 x 107
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35 Which is ordered from least to greatest?
A I, II, III, IV
B IV, III, I, II
C I, IV, II, III
D III, IV, I, II
I. 1.0 x 103
II. 5.0 x 103
III. 8.3 x 106
IV. 9.5 x 106
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36 Which is ordered from least to greatest?
A I, II, III, IV
B IV, III, I, II
C I, IV, II, III
D III, IV, I, II
I. 2.5 x 10-3
II. 5.0 x 10-3
III. 9.2 x 10-6
IV. 4.2 x 10-6
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Multiplying Numbers in Scientific Notation
Multiplying with scientific notation requires at least three, and sometimes four, steps.
1. Multiply the coefficients
2. Multiply the powers of ten
3. Combine those results
4. Put in proper form
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6.0 x 2.5 = 15
104 x 102 = 106
15 x 106
1.5 x 107
1. Multiply the coefficients
2. Multiply the powers of ten
3. Combine those results
4. Put in proper form
Evaluate: (6.0 x 104)(2.5 x 102)
Multiplying Numbers in Scientific Notation
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4.8 x 9.0 = 43.2
106 x 10-8 = 10-2
43.2 x 10-2
4.32 x 10-1
1. Multiply the coefficients
2. Multiply the powers of ten
3. Combine those results
4. Put in proper form
Evaluate: (4.80 x 106)(9.0 x 10-8)
Multiplying Numbers in Scientific Notation
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37 Evaluate (2.0 x 10-4)(4.0 x 107). Express the result in scientific notation.
A 8.0 x 1011
B 8.0 x 103
C 5.0 x 103
D 5.0 x 1011 E 7.68 x 10-28 F 7.68 x 10-28
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38 Evaluate (5.0 x 106)(7.0 x 107)
A 3.5 x 1013 B 3.5 x 1014 C 3.5 x 101 D 3.5 x 10-1 E 7.1 x 1013 F 7.1 x 101
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39 Evaluate (6.0 x 102)(2.0 x 103)
A 1.2 x 106
B 1.2 x 101
C 1.2 x 105
D 3.0 x 10-1
E 3.0 x 105
F 3.0 x 101
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40 Evaluate (1.2 x 10-6)(2.5 x 103). Express the result in scientific notation.
A 3 x 103
B 3 x 10-3
C 30 x 10-3
D 0.3 x 10-18
E 30 x 1018
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41 Evaluate (1.1 x 104)(3.4 x 106). Express the result in scientific notation.
A 3.74 x 1024
B 3.74 x 1010
C 4.5 x 1024
D 4.5 x 1010
E 37.4 x 1024
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42 Evaluate (3.3 x 104)(9.6 x 103). Express the result in scientific notation.
A 31.68 x 107
B 3.168 x 108
C 3.2 x 107
D 32 x 108
E 30 x 107
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43 Evaluate (2.2 x 10-5)(4.6 x 10-4). Express the result in scientific notation.
A 10.12 x 10-20
B 10.12 x 10-9
C 1.012 x 10-10
D 1.012 x 10-9
E 1.012 x 10-8
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Dividing Numbers in Scientific Notation
Dividing with scientific notation is just like multiplying.
1. Divide the coefficients
2. Divide the powers of ten
3. Combine those results
4. Put in proper form
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Division with Scientific Notation
5.4 ÷ 9.0 = 0.6
106 ÷ 102 = 104
0.6 x 104
6.0 x 103
1. Multiply the coefficients
2. Multiply the powers of ten
3. Combine those results
4. Put in proper form
Evaluate: 9.0 x 1025.4 x 106
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Division with Scientific Notation
4.4 ÷ 1.1 = 4.0
106 ÷ 10-3 = 109
4.0 x 109
1. Multiply the coefficients
2. Multiply the powers of ten
3. Combine those results
4. Put in proper form
Evaluate: 1.1 x 10-34.4 x 106
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44 Evaluate 4.16 x 10-9
5.2 x 10-5 Express the result in scientific notation.
A 0.8 x 10-4
B 0.8 x 10-14
C 0.8 x 10-5
D 8 x 10-4
E 8 x 10-5
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45 Evaluate 7.6 x 10-2
4 x 10-4 Express the result in scientific notation.
A 1.9 x 10-2
B 1.9 x 10-6
C 1.9 x 102
D 1.9 x 10-8
E 1.9 x 108
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46 Evaluate 8.2 x 103
2 x 107 Express the result in scientific notation.
A 4.1 x 10-10
B 4.1 x 104
C 4.1 x 10-4
D 4.1 x 1021
E 4.1 x 1010
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47 Evaluate 3.2 x 10-2
6.4 x 10-4 Express the result in scientific notation.
A .5 x 10-6
B .5 x 10-2
C .5 x 102
D 5 x 101
E 5 x 103
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48 The point on a pin has a diameter of approximately 1 x 10-4 meters. If an atom has a diameter of 2 x 10-10 meters, about how many atoms could fit across the diameter of the point of a pin?
A 50,000 B 500,000C 2,000,000D 5,000,000
Question from ADP Algebra I End-of-Course Practice Test
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Addition and Subtraction with Scientific Notation
Numbers in scientific notation can only be added or subtracted if they have the same exponents.
If needed, an intermediary step is to rewrite one of the numbers so it has the same exponent as the other.
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Addition and Subtraction
This is the simplest example of addition
4.0 x 103 + 5.3 x 103 =
Since the exponents are the same (3), just add the coefficients.
4.0 x 103 + 5.3 x 103 = 9.3 x 103
This just says that if you add 4.0 thousand and 5.3 thousand to get 9.3 thousand.
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Addition and Subtraction
This problem is slightly more difficult because you need to add one extra step at the end.
8.0 x 103 + 5.3 x 103 =
Since the exponents are the same (3), just add the coefficients.
4.0 x 103 + 5.3 x 103 = 13.3 x 103
But that is not proper form, since 13.3 > 10; it should be written as 1.33 x 104
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Addition and Subtraction 8.0 x 104 + 5.3 x 103 =
This requires an extra step at the beginning because the exponents are different. We have to either convert the first number to 80 x 103 or the second one to 0.53 x 104.
The latter approach saves us a step at the end.
8.0 x 104 + 0.53 x 104 = 8.53 x 104
Once both numbers had the same exponents, we just add the coefficient.
Note that when we made the exponent 1 bigger, that's makes the number 10x bigger; we had to make the coefficient 1/10 as large to keep the number the same.
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49 The sum of 5.6 x 103 and 2.4 x 103 is
A 8.0 x 103
B 8.0 x 106
C 8.0 x 10-3
D 8.53 x 103
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50 8.0 x 103 minus 2.0 x 103 is
A 6.0 x 10-3
B 6.0 x 100
C 6.0 x 103
D 7.8 x 103
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51 7.0 x 103 plus 2.0 x 102 is
A 9.0 x 103
B 9.0 x 105
C 7.2 x 103
D 7.2 x 102
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52 3.5 x 105 plus 7.8 x 105 is
A 11.3 x 105
B 1.13 x 104
C 1.13 x 106
D 11.3 x 1010
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Roots
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The symbol for taking a square root is , it is a radical sign.The square root cancels out the square. There is no real square root of a negative number.
is not real (42=16 and (-4)2=16)
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53 What is 1 ?
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54 What is ?
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55 What is ?
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To take the square root of a variable rewrite its exponent as the square of a power.
Square roots need to be positive answers. Even powered answered, like above, are positive even if the variables negative. The same cannot be said if the answer has an odd power. When you take a square root an the answer has an odd power, put the answer inside of absolute value signs.
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62
A
B
C
D no real solution
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63
A
B
C
D no real solution
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64
A
B
C
D no real solution
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65
A
B
C
D no real solution
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66 Evaluate
A B
C D No Real Solution
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67 Evaluate
A B
C D No Real Solution
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68 Evaluate
A B
C D No Real Solution
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Q: If a square root cancels a square, what cancels a cube?
A: A cube root.
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The volume (V) of a cube is found by cubing its side length (s).
V = s3
V = s3
V = 43 = 4 4 4
V = 64 cubic units or 64 units3
4 units
The volume (V) of a cube is labeled as cubic units, or units3, because to find the volume, you need to cube its side.
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A cube with sides 3 units would have a volume of 27 u3 because 33=27.
If a cube has an volume of 64 u3 what is the length of one side?
Need to find a number when multiplied by itself three times will equal 64.
4 4 4 = 64, so 4 units is the length of a side.
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75 Simplify
A B
C D not possible
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76 Simplify
A B
C D not possible
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79 Which of the following is not a step in simplifying
A
B
C
D
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In general, and absolute value signs are needed if n is even and the variable has an odd powered answer.
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82 Simplify
A
B
C
D
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83 Simplify
A
B
C
D
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85 Simplify
A
B
C
D
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86 Simplify
A
B
C
D
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87 Simplify
A
B
C
D
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88 If the nth root of a radicand is , which of the following is always true?
A No absolute value signs are ever needed.
B Absolute value signs will always be needed.
C Absolute value signs will be needed if j is negative.
D Absolute value signs are needed if n is an even index.
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Simplifying Radicals
is said to be a rational answer because their is a perfect square that equals the radicand.
If a radicand doesn't have a perfect square that equals it, the root is said to be irrational.
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The square root of the following numbers is rational or irrational?
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The commonly excepted form of a radical is called the "simplified form".
To simplify a non-perfect square, start by breaking the radicand into factors and then breaking the factors into factors and so on until there
only prime numbers are left. this is called the prime factorization.
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89 Which of the following is the prime factorization of 24?
A 3(8)
B 4(6)
C 2(2)(2)(3)
D 2(2)(2)(3)(3)
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90 Which of the following is the prime factorization of 72?
A 9(8)
B 2(2)(2)(2)(6)
C 2(2)(2)(3)
D 2(2)(2)(3)(3)
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91 Which of the following is the prime factorization of 12?
A 3(4)
B 2(6)
C 2(2)(2)(3)
D 2(2)(3)
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92 Which of the following is the prime factorization of 24 rewritten as powers of factors?
A
B
C
D
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93 Which of the following is the prime factorization of 72 rewritten as powers of factors?
A
B
C
D
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94 Simplify
A
B
C
D already in simplified form
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95 Simplify
A
B
C
D already in simplified form
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96 Simplify
A
B
C
D already in simplified form
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97 Simplify
A
B
C
D already in simplified form
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98 Which of the following does not have an irrational simplified form?
A
B
C
D
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Simplifying Roots of Variables
Divide the index into the exponent. The number of times the index goes into the exponent becomes the power on the outside of the
radical and the remainder is the power of the radicand.
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Simplifying Roots of VariablesWhat about the absolute value signs?
An Absolute Value sign is needed if the index is even, the starting power of the variable is even and the answer is an odd power on the outside.
Examples of when absolute values are needed:
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101 Simplify
A
B
C
D
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102 Simplify
A
B
C
D
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Operations with Radicals
To add and subtract radicals they must be like terms.
Radicals are like terms if they have the same radicands and the same indexes.
Like Terms Unlike Terms
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103 Identify all of the pairs of like terms
A
B
C
D
E
F
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To add or subtract radicals, only the coefficients of the like terms are combined.
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105 Simplify
A
B
C
D Already Simplified
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107 Simplify
A
B
C
D Already Simplified
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108 Simplify
A
B
C
D Already Simplified
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Some irrational radicals will not be like terms, but can be simplified. In theses cases, simplify then check for like terms.
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109 Simplify
A
B
C
D Already in simplest form
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112 Simplify
A
B
C
D Already in simplest form
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113 Which of the following expressions does not equal the other 3 expressions?
A
B
C
D
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114 Multiply
A
B
C
D
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Multiplying Square Roots
After multiplying, check to see if radicand can be simplified.
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115 Simplify
A
B
C
D
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116 Simplify
A
B
C
D
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117 Simplify
A
B
C
D
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118 Simplify
A
B
C
D
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Multiplying Polynomials Involving Radicals 1) Follow the rules for distribution. 2)Be sure to simplify radicals when possible and combine like terms.
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119 Multiply and write in simplest form:
A
B
C
D
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120 Multiply and write in simplest form:
A
B
C
D
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121 Multiply and write in simplest form:
A
B
C
D
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122 Multiply and write in simplest form:
A
B
C
D
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123 Multiply and write in simplest form:
A
B
C
D
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Rationalizing the Denominator
Mathematicians don't like radicals in the denominators of fractions. When there is one, the denominator is said to be irrational. The method used to rid the denominator is termed "rationalizing the
denominator".Which of these has a rational denominator?
RationalDenominator
IrrationalDenominator
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If a denominator needs to be rationalized, start by finding its conjugate.A conjugate is another polynomial that when the conjugate and the denominator are multiplied, no more irrational term.
The conjugate for a monomial with a square root is the same square root.Example has a conjugate of . Why? Because
The conjugate of a binomial with square roots is the opposite operation between the terms.Example has a conjugate of . Why? Because
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Can you find a pattern for when a binomial is multiplied by its conjugate?Example Example Example
Do you see a pattern that let's us go from line 1 to line 3 directly?
(term 1)2 - (term 2)2
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124 What is conjugate of ?
A
B
C
D
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125 What is conjugate of ?
A
B
C
D
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127 What is conjugate of ?
A
B
C
D
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The goal is to rationalize the denominator without changing the value of the fraction. To do this multiply the numerator and denominator by the same exact value.
Examples:
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Rationalize the Denominator:
The original x in the radicand had an odd
power.
Why no absolute value signs?
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Rationalize the Denominator:
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129 Simplify
A
B
C
D Already simplified
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130 Simplify
A
B
C
D Already simplified
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131 Simplify
A
B
C
D Already simplified
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Rationalizing nth roots of monomials
Remember that , given an nth root in the denominator, you will need to find the conjugate that makes the radicand to the nth power.
Examples:
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134 Rationalize
A
B
C
D
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136 Rationalize
A
B
CD
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Rational Exponents
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Rational Exponents, or exponents that are fractions, is another way to write a radical.
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Rewrite each radical as a rational exponent in the lowest terms.
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140 Find the simplified expression that is equivalent to:
A
B
C
D
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141 Find the simplified expression that is equivalent to:
A
B
C
D
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142 Find the simplified expression that is equivalent to:
A
B
C
D
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143 Find the simplified expression that is equivalent to:
A
B
C
D
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144 Simplify
A
B
C
D
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145 Simplify
A
B
C
D
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147 Simplify
A
B
C
D
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