Review of Radicals

MATH 017

Intermediate Algebra

S. Rook

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Overview

• Section 7.1 in the textbook– Find square roots– Approximate roots– Find cube roots– Find nth roots

Find Square Roots

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Find Square Roots

• Should be a review for numbers means “what number multiplied by itself gives you a”?

• What about the square root of a negative number?– Suppose we want to evaluate

• What number multiplied by itself gives you -4?There is none because the product of two numbers with the

same sign is always positive!

• Therefore, the square root of a negative number does NOT exist in the real number system

a

4

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Find Square Roots (Continued)

• Slightly different for variables– Consider

xxxx

)()( xxxx

2x

4x

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Find Square Roots (Continued)

• Thus: if a is divisible by 2• Perfect squares

– Should have the first ten perfect squares memorized

2aa xx

x x2 x x2 x x2 x X2

1 1 4 16 7 49 10 100

2 4 5 25 8 64

3 9 6 36 9 81

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Find Square Roots (Example)

Ex 1: Evaluate 82 4,,144,16,0 yx

Approximate Roots

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Approximate Roots

• Most square roots will not evaluate to integers, but to irrational numbers

• Can approximate by “squeezing” the root between two perfect squares

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Approximate Roots (Example)

Ex 2: Approximate and then evaluate it using a calculator

51

Find Cube Roots

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Find Cube Roots

• Should be a review for numbers means “what number multiplied by itself three times gives you a”?

• What about the cube root of a negative number?– Suppose we wish to evaluate

• What number multiplied by itself three times gives you -8?

-2

• Therefore, the cube root of a negative number exists in the real number system because the product of three negatives is negative

3 a

3 8

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Find Cube Roots (Continued)

• Slightly different for variables– Consider

3 xxxxxx

3 )()( xxxxxx

2x

3 6x

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Find Cube Roots (Continued)

• Thus: if a is divisible by 3• Perfect cubes

– Should have the first five perfect cubes memorized

33 aa xx

x x3 x x3

1 1 4 64

2 8 5 125

3 27

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Find Cube Roots (Example)

Ex 3: Evaluate 3 123 333 ,27,125,0 ba

Find nth Roots

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Find nth Roots

• Should be a review for numbers means “what number multiplied by itself n times gives you a”?

• What about the nth root of a negative number?– Already saw that the cube root of a negative number exists

in the real number system exists while the square root of a negative number does not

– Can extend this to the general case• The product of an even number of negatives is positive

– Therefore, the even root of a negative number does NOT exist in the real number system

• The product of an odd number of negatives is negative– Therefore, the odd root of a negative number DOES exist in the

real number system

n a

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Find nth Roots (Continued)

• Slightly different for variables– Consider

4 xxxxxxxx

4 )()( xxxxxxxx

2x

4 8x

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Find nth Roots (Continued)

• Thus: if a is divisible by nnan a xx

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Find nth Roots (Example)

Ex 4: Evaluate 5 104 1644 32,,81,16 wz

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Summary

• After studying these slides, you should know how to do the following:– Evaluate square, cube, and nth roots– Approximate a root