Exponent and Radical Rules (6.1, 6.2) Day 20

Name: ___________________________________________________ Block: ______________

Topic Definition/Rule Example(s)

Multiplication xa ⋅ xb = xa+b

Power to a Power xa( )b = xab

Power of a Product

ab( )n =anbn

Zero

Exponents x0 =1

Division

xa

xb = xa−b

Power of a Quotient

xy

⎛

⎝⎜

⎞

⎠⎟

n= xn

yn

Simplifying

1.

2.

3.

Simplifying Exponents

Step Method Example

1 Label all unlabeled exponents “1”

2 Take the reciprocal of the fraction and make the outside exponent positive.

3 Get rid of any inside parentheses.

4 Reduce any fractional coefficients.

5 Move all negatives either up or down.

Make the exponents positive.

6 Combine all like bases.

7 Distribute the power to all exponents.

Use properties of exponents to simplify the following expressions.

f.

g. h. 121/8 125/6 =

i. (51/3 x1/4)3 = j. (26 46)-1/6 =

k. = l.

RATIONAL EXPONENTS

QUESTION: What is the square root of a number? Determine what number fits into the . a. 5 i 5 = 5 is the _____________ of 25 b. 10 i 10 = 10 is the _____________ of 100 c. x i x = _____ is the square root of ______ d. x 3 i x 3 = _____ is the square root of ______ e. x 9 i x 9 = _____ is the square root of ______ QUESTION: What does the square root of x mean?

x i x = x1 is the square root of x. RULE: __________________________

FRACTIONAL EXPONENT RULE:

For any real number a and integers n and m: _____________________________

Examples:

a. 161/2 = = 4 b. 271/3 = = 3

c. (-8)1/3 = = -2 d. (16)1/4 = = 2

RATIONAL EXPONENTS

Exponent (numerator)

x1/2

Base (x) Root (denominator)

RADICALS

Index

Radicand

ExponentialNotation

RadicalNotation

x1/2

x2/3

x3/4

am/n

a-m/n

RULE:

EXAMPLES:

Evaluate each of the following without the use of a calculator!

1. 2. 3. 4.

5. 6. 7. 8.

9. 10. 11. 12.

RULE:

EXAMPLES:

Evaluate each of the following without the use of a calculator!

1. 2. 3. 4.

5. 6. 7. 8.

9. 10. 11. 12.

A negative exponent was slipped into that last problem! How did you deal with it?

RULE:

EXAMPLES:

Evaluate each of the following without the use of a calculator!

1. 2. 3. 4.

5. 6. 7. 8.

9. 10. 11. 12.

Mad Math Minute!

Name: __________________________________ Date: __________

1. x2

3 = ______ 2.

x2( )3 = ______ 3.

x4( )7 = ______

4.

x3( )4 = _____ 5. x13 = ______ 6. 100

12 = ______

7. y13 = ______ 8.

x 34⎛

⎝⎜⎞⎠⎟

= ______ 9. x3

2 = ______

10. 4912 = ______ 11. x

15 = ______ 12.

x 4⎛

⎝⎜⎞⎠⎟

= ______

13. x12 = ______ 14. 16

12 = ____ 15.

x 35⎛

⎝⎜⎞⎠⎟

= ______

16. x51 = ______ 17.

x15⎛

⎝⎞⎠

3 = ______ 18. 4

32 = ______

RADICALS

Warm Up Simplify the following square root and cube root expressions

1.

2.

3.

4.

EXAMPLE ONE Simplifying square roots with variables

a) b)

EXAMPLE TWO Rationalizing the denominator

a)

5

63 b)

7

1+ 2 5

EXAMPLE THREE Multiplying Radical Expressions

c) d)

EXAMPLE FOUR Adding and Subtracting Radical Expressions

a)

b)

c)

PRACTICE

1. 2. 3. x11

4. 5.

6.

Simplify using the exponent rules. (No decimals, keep as fractions)

1.

2. 3.

4.

5. (a3b‐6)(a2b0)6.

7. 8. 9.

1. 5. 6.

7. 9.

10.

12. 243 − 33

15. 6 323 − 5 43 4.

2. 13.

53 + 7

14.

21− 4 3

EXPONENT PROPERTIES

EXAMPLE FOUR Re-write the radical expression in exponential form.

a) b) c) d)

EXAMPLE FIVE Simplify the expression without using a calculator.

a)

b)

c)

d)