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Algebra practice part 4

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E. Exponents

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Positive exponents

(convention)

(x any number,n positive integer)

3-rd power of 4, 4: base, 3: exponent

Examples:

In general:

Exercises:

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Negative exponents

(x any non-zero number,n positive integer)

x-1 is the inverse of x

Examples:

In general:

Exercises:

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Radicals

?3 = 8

• 23=8: 2 is the 3-rd root (cubic root) of 8

• the 3-rd root of 8 is denoted by

i.e.

Example:

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Radicals

?3 = –8

• (–2)3=8: –2 is the 3-rd root of –8

• the 3-rd root of 8 is denoted by

i.e.

Example:

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Radicals

?4 = 16

• 24=16: 2 is a 4-th root of 16

• (–2)4=16: also –2 is a 4-th root of 16

• 16 has two 4-th roots: 2 and -2

• positive 4-th root of 16 is denoted by

i.e.

• it follows that the negative 4-th root of 16 is

given by

i.e.

Example:

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Radicals

?4 = –16

• no numbers whose 4-th power equals –16

• –16 has no 4-th root

Example:

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Radicals

• 16 has two 4-th roots: and

this is a typical example of the case of an even root of a positive number

• –16 has no 4-th roots

this is a typical example of the case of an even root of a negative number

• 8 has one 3-rd root:

this is a typical example of the case of an odd root of a positive number

• –8 has one 3-rd root:

this is a typical example of the case of an odd root of a negative number

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Radicals: remarks

• 3-rd roots are cubic roots

• 2-nd roots are square roots:

• for any positive integer n:

• in many cases roots have to be calculated usingthe calculator:

♦

♦ …

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Fractional-exponent-notation for roots

(x any stricly positive number,n positive integer)

In general:

Exercises:

Example:

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More general fractions as exponent

(x any strictly positive number,z integer, n positive integer)

Examples:

stands for , i.e.

In general:

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Irrational exponents14

Product of powers with same base

x3 ⋅ x4 can be written in a simpler form :

In general (real exponents and positive bases):

Example:

Exercise:

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Quotient of powers with same base

x5 / x3 can be written in a simpler form :

In general (real exponents and positive bases):

Example:

Exercise:

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Power of a power

(x3)2 can be written in a simpler form :

In general (real exponents and positive bases):

Example:

Exercise:

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Power of a power: a special case

ONLY for positive x-values!

rational exponents forpositive bases only, not

valid for x= –2

18 Product of powers with same exponentPower of a product

x3⋅y3 can be written in a different form:

(x⋅y)3 can be written in a different form

Example:

Exercise:

In general (real exponents and positive bases):

19Quotient of powers with same exponentPower of a quotient

x3/y3 can be written in a different form:Example:

Exercise:

In general (real exponents and positive bases):

20 Sum of powers with same exponentPower of a sum

(x+y)r can NOT be written in a simpler form:

Examples:

In general:

=

==

=

21 Sum of powers with same exponentPower of a sum

In general:

Further examples:

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Rules for exponents: summary

for all real exponents and positive bases:

same base:

same exponent:

power of a power:

applied to (square) roots:

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Equations with powers: example 1

The volume of a cube with side x is given by V=x3.

1. Find the volume of a cube having side 4 cm.

2. What is the side of a cube having volume 729 cm3?

3. A first cube has side 3 cm. Find the side of a second cube, whose volume is the double of the volume of the first one.

Answers:

1. 64 cm3

2. solving x3=729 gives x=7291/3=9 (cm)

3. solving x3=2⋅33 gives x=3⋅21/3=3.77…≈3.8 (cm)

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Write y in terms of x if y3 = 5⋅x2.

y3 = 5⋅x2

y = 51/3⋅(x2)1/3

Answer: y = 51/3⋅x2/3

( )1/3 ( )1/3

Equations with powers: example 2

we have to get rid of the exponent 3

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E. Exponents

Handbook

Chapter 0: Review of Algebra

0.3 Exponents and Radicals

(except: rationalizing denominators, i.e. example 3, example 6.c, problems 59-68)