Algebra practice part 4 E. Exponents 3 4 Positive exponents - - PowerPoint PPT Presentation

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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 10

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 using

the 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 exponents

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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 for positive bases only, not valid for x= –2

18 Product of powers with same exponent

Power 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):

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Quotient of powers with same exponent Power 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 exponent

Power of a sum

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

Examples: In general: = = = =

21 Sum of powers with same exponent

Power 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

• f 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)