chapter 4 exponents and polynomials. the rules of exponents chapter 4.1
TRANSCRIPT
Chapter 4
Exponents and Polynomials
The Rules of Exponents
Chapter 4.1
The Product Rule
xa • xb = xa + b
1. Multiply.
a.
a12
a • a • a • a • a • a • a • a • a • a • a • a
a7 + 5
• a5a7
Count how many you are multiplying.Just add the exponents.
1. Multiply.
b. w10 • w
w10 + 1
w11
Add the exponents.
2. Simplify, if possible.
a. x3 • x9
x3 + 9
x12
Same base, add the exponents.
2. Simplify, if possible.
b. 37 • 34
37 + 4
311
Same base, add the exponents.
2. Simplify, if possible.
c. a3 • b2
a3b2
Different bases, can’t use the product rule.
3. Multiply.
a. (-a8)(a4)
-
a8 + 4
a12
Multiply the coefficients.Add the exponents.
(-1)(1)
3. Multiply.
b. (3y2)(-2y3)
-6
y2 + 3
y5
Multiply the coefficients.Add the exponents.
(3)(-2)
3. Multiply.
c. (-4x3)(-5x2)
20
x3 + 2
x5
Multiply the coefficients.Add the exponents.
(-4)(-5)
(
(2)(-)(6)
-3
(x1 + 2 + 1) (y1 + 1 + 3)1
21
3
4. Multiply.
x4y5
Multiply and simplify the coefficients.Add the exponents for x and then for y.
2 )(yx 6)(yx2- y3x )
xa • xb = xa + b
The Product Rule
xb
xa
= xa – b if a > b
xb
xa
= if b > axb – a
1
xa
xa
= x0 = 1
1.
2.
3.
The Quotient Rules
The Quotient Rules
xb
xa
= xa – b if a > b1.
a.
106
1013
107
10 • 10 • 10 • 10 • 10 • 10 • 10 • 10 • 10 • 10 • 10 • 10 • 1010 • 10 • 10 • 10 • 10 • 10 • 10
1013 – 7
5. Divide.
Count how many are crossed out.Just subtract the exponents.
b.
x11 – 1
x10
xx11
5. Divide.
Higher exponent in numerator.Subtract the exponents.
c.
y18 – 8
y10
y8
y18
5. Divide.
Higher exponent in numerator.Subtract the exponents.
The Quotient Rules
xb
xa
= xa – b if a > b
xb
xa
= if b > axb – a
1
1.
2.
a.c4
c3
c4 – 3
1
c1
c • c • cc • c • c • c
6. Divide.
The higher exponent is in the denominator.
b.1056
1031
1056 – 31
1
1025
1
6. Divide.
The higher exponent is in the denominator.
c.z21
z15
z21 – 15
1
z6
1
6. Divide.
The higher exponent is in the denominator.
a.-21-7
31
3x2
1
x9 – 7
7. Divide.
Simplify.
x7
x9
The higher exponent is in the denominator.
b.-315
1-5 x11 – 4
7. Divide.
Simplify.
x11
x4
The higher exponent is in the numerator.
-5x7
c.4623
21
x9 – 8
7. Divide.
Simplify.
x8
x9
The higher exponent is in the denominator.
2x1
a.y10
x7
x7
yx7
y10 – 9
8. Divide.
y9
The higher exponent is in the denominator.
Can’t simplify.
b.-2412
2-1
2y2
-x2
x5 – 3
y8 – 6
8. Divide.
x5y6
x3y8
The higher exponent is in the numerator.
Simplify.
The higher exponent is in the denominator.
The Quotient Rules
xb
xa
= xa – b if a > b
xb
xa
= if b > axb – a
1
xa
xa
= x0 = 1
1.
2.
3.
a.
1
107107
= 100
9. Divide.
Same exponent.
b.1512
54
9. Divide.
a4
a4
Simplify.Same exponent.
a.28-20
7-5
7c-5b
b8 – 7
c5 – 4
10. Divide.
a3b8c4
a3b7c5
The higher exponent is in the numerator.
Simplify.
The higher exponent is in the denominator.
Same exponent.
x0b.
105
21
x4 y8 – 6
10. Divide.
y6
x4 y8
The higher exponent is in the denominator.
Simplify.
0 exponent.
2x4y2
1
16a5b7
(
-18a3
16b9
-98a2
b2
11. Simplify.
)(ab5-6 3a2b4 )
Simplify.
Multiply.
Subtract the exponents.
Add the exponents in the numerator.
a5b7
The Quotient Rules
xb
xa
= xa – b if a > b
xb
xa
= if b > axb – a
1
xa
xa
= x0 = 1
1.
2.
3.
The Power Rules
(xa)b = xa • b
(xayb)c = xa • c
( )yb
xac xa • c
yb • c=
yb • c
a. (a4)3
a4 • 3
a12
(a4)(a4)(a4)
12. Simplify.
Can write it three times.
Add 4 three times or multiply the exponents.
b. (105)2
105 • 2
1010
12. Simplify.
Multiply the exponents.
c. (-1)15
-1
12. Simplify.
Multiply -1 an odd number (15) of times.
a. (3xy)3
(3)3
27
x1 • 3 y1 • 3
13. Simplify.
Multiply the exponents.
Keep 3 in the parentheses.
Evaluate each.
x3y3
y1 • 37 z1 • 37
13. Simplify.
Multiply the exponents.
Evaluate each.
y37 z37
b. (yz)37
(-3)2
9
x3 • 2
13. Simplify.
Multiply the exponents.
Keep -3 in the parentheses.
Evaluate each.
x6
c. (-3x3)2
a.5x
( )3
(5)3
x3
125x3
14. Simplify.
Keep 5 in the parentheses.
Multiply the exponents.
Evaluate.
14. Simplify.
b. ((4a)2
ab)6
Multiply exponents.
Evaluate.
Use quotient rule and subtract exponents.
16a2
a6b6
16a4b6
( )5
4-2
15. Simplify.
x3y0zxz2
Simplify and use quotient rules.
Work inside parentheses.
Use power rule and evaluate.
2-1x2
z( )5
32- x10
z5
The Power Rules
(xa)b = xa • b
(xayb)c = xa • c
( )yb
xac xa • c
yb • c=
yb • c
The Rules of Exponents
Chapter 4.1