Exponents & Roots

Remember, when you have an exponent that is a fraction, its power over root.

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​Rational Exponents

power

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root

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exponential form​

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radical form​

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We don't write the "2" because it's implied!​

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​Dr. Tom Giles

Multiple Choice

Which of the following is correct?

• $$\frac{Power}{Root}$$

• $$\frac{Root}{Power}$$

Multiple Choice

Which of the following orders is correct when simplifying expressions with rational exponents?

• Take the root, then the power.

• Take the power, then the root.

Multiple Choice

Which of the following is in radical form?

• $$\sqrt[4]{5^3}$$  

• $$5^{\frac{3}{2}}$$  

Multiple Choice

Which of the following is in exponential form?

• $$\sqrt[4]{8^{ }}$$  

• $$8^{\frac{1}{4}}$$  

Multiple Choice

Write the following expression in radical form: $$27^{\frac{2}{3}}$$  

• $$\sqrt[2]{27^3}$$  

• $$\sqrt[]{27^3}$$  

• $$\sqrt[3]{27^2}$$  

• $$\sqrt[]{27^2}$$  

Multiple Choice

Write the following expression in radical form: $$4^{\frac{5}{2}}$$  

• $$\sqrt[]{4^5}$$  

• $$\sqrt[5]{4^2}$$  

• $$\sqrt[5]{4}$$  

Multiple Choice

Write the following expression in radical form: $$81^{\frac{1}{2}}$$  

• $$\sqrt[]{81}$$  

• $$\sqrt[]{81^2}$$  

• $$\sqrt[1]{81^2}$$  

• $$81^2$$  

Match

Match each expression to it's radical form:

• $$\sqrt[]{25^5}$$  

• $$\sqrt[5]{25^2}$$  

• $$\sqrt[]{5^{25}}$$  

• $$\sqrt[25]{5^2}$$  

 

 

 

 

• $$25^{\frac{5}{2}}$$  

• $$25^{\frac{2}{5}}$$  

• $$5^{\frac{25}{2}}$$  

• $$5^{\frac{2}{25}}$$  

For example:

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​When simplifying an expression with a rational exponent, take the root first!

​-Take the 5th root of 32 first

​-Now take the 2nd power

Multiple Choice

When simplifying an expression with a rational exponent, first we ...

• take the root

• take the power

Exponent Properties

When multiplying powers add the exponents

Multiple Choice

Simplify the following expression using exponential notation.

$$2^2\cdot2^4$$  

• $$2^8$$  

• $$2^6$$  

• $$4^6$$  

• $$4^8$$  

Exponent Properties

When you have a power of a power, multiply the exponents

Multiple Choice

(x³)⁴

• x¹²

• x^-1

• x⁷

• x⁰

Exponent Properties

When you have a power of a product, distribute the exponent.

Multiple Choice

Simplify: (3x²)³

• 27x⁶
• 9x⁶
• 3x⁶
• 9x⁵

Exponent Properties

When you have a quotient of powers, subtract the exponent.

Multiple Choice

Write the following expression in radical form: $$64^{\frac{1}{3}}$$  

• $$\sqrt[]{64^3}$$   

• $$\sqrt[1]{64^3}$$   

• $$\sqrt[3]{64}$$  

Multiple Choice

$$Simplify\ \ \frac{x^8}{x^2}$$  

• $$x^4$$  

• $$x^6$$  

• $$6x$$  

Exponent Properties

When you have a negative exponent, take the reciprocal to make positive.

Multiple Choice

Rewrite using a positive exponent.
7^-10

• ¹/_7¹⁰
• 7¹⁰
• ¹/_7^-10
• -70

Multiple Choice

Simplify the expression: 
c⁴⋅c³=

• ^c12
• c⁴⁺³
• c⁷

Multiple Choice

Which of the following steps would you take when simplifying the expression $$\sqrt[]{25^3}$$

• Raise 25 to the 3rd power, then take the square root.

• Take the square root of 25, then raise it to the 3rd power.

Multiple Choice

Simplify the expression: 3x²⋅x²=

• 3x
• 3x²⁺²
• 3x⁴

Multiple Choice

According to exponent rules, when we raise a power to another exponent we _______ the exponents.

• add
• subtract
• multiply
• divide

Multiple Choice

Simplify 4^-2= 

• 1/16
• -16
• 1/4
• -42

Multiple Choice

Simplify

Multiple Choice

Simplify

Multiple Choice

Simplify

Multiple Choice

Anything raised to a power of zero is always: 

• 0

• 1

• itself

• negative

Multiple Choice

(5³x²y⁴)⁰

• 5xy

• 1

• 0

• 5

Match

Match the following:

• 16

• 4

• 64

• 9

• 81

 

 

 

 

 

• $$\sqrt[3]{8^4}$$  

• $$\sqrt[4]{16^2}$$  

• $$\sqrt[]{16^3}$$  

• $$\sqrt[4]{81^2}$$  

• $$\sqrt[3]{27^4}$$  

Multiple Choice

Simplify

(2a²b⁴z)(6a³b²z⁵)

• 8a⁵b⁶z⁶

• 12a⁶b⁸z⁵

• 12a⁵b⁶z⁶

• 8a⁶b⁸z⁵

For example:

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If the expression is written in exponential form, convert it to radical form in order to simplify it.

exponential form​

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radical form​

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2.) Take the "root" ​

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1.) Write in radical form​

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3.) Take the "power" ​

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Multiple Choice

(6x²)(-3x⁵)

• 18x⁷

• -18x⁷

• 18x⁷

• 3x⁷

Reorder

Reorder in the order when simplifying $$8^{\frac{2}{3}}$$ :

• $$8^{\frac{2}{3}}$$  

• $$=\sqrt[3]{8^2}$$  

• $$=2^2$$  

• $$=4$$  

1

2

3

4

• 1
• 2
• 3
• 4

Multiple Choice

When dividing powers with the same base, you _______________ the exponents.

• Add

• Subtract

• Multiply

• Divide

Multiple Choice

Simplify: $$8^{\frac{4}{3}}$$  

• 2

• 8

• 16

• 4

Multiple Choice

Simplify

• x⁴/3

• 3x¹⁰

• 36x¹⁰

• 3x⁴

Multiple Choice

Simplify: $$16^{\frac{1}{2}}$$  

• 4

• 8

• 32

• 16

Multiple Choice

• 2 / x²

• -2x²

• 2x²

• 2x¹⁴

Multiple Choice

Simplify.  

• A

• B

• C

• D

Multiple Choice

• A

• B

• C

• D

Multiple Choice

Multiple Choice

Multiple Choice

Simplify the expression shown. Your answer should contain only positive exponents.

• 16a¹²b¹⁴

Radical

Expressions​

• The opposite of a root is an exponent!

  ​To simplify SQUARE ROOTS:

  • Write the radicand as a product of numbers, one of which is a PERFECT SQUARE.

  • Find the square root of the perfect square.​

Radical Expressions

Basics/Square Roots

Multiple Choice

$$\sqrt[]{36}$$  

• 9

• 6

• 4

• 8

• NOT ALL SQUARE ROOTS ARE PERFECT!

  • When this happens you should follow the same process ,but you will need to leave any remaining products that cannot come out of the radical​ underneath the root.

Radical Expressions

Basics/Square Roots

Multiple Choice

$$\sqrt[]{18}$$  

• $$3\sqrt[]{2}$$  

• $$2\sqrt[]{9}$$  

• $$2\sqrt[]{3}$$  

• $$\sqrt[]{9}$$  

​Exponent

Multiple Choice

$$\sqrt[3]{54}$$  

• $$3\sqrt[3]{18}$$  

• $$2\sqrt[3]{3}$$  

• $$3\sqrt[3]{2}$$  

• $$18\sqrt[3]{3}$$  

Multiple Choice

$$\sqrt[3]{64}$$  

• 32

• 4

• 8

• 2

Open Ended

What does B.I.T.E stand for?

Fractional Exponents can be written as radical expressions!

Bottom

Index

Top

Exponent​

Radical Expressions

Fractional Exponents

Multiple Choice

$$13^{\frac{1}{4}}$$  

• $$\sqrt[]{13}$$  

• $$\sqrt[4]{13}$$  

• $$\left(\sqrt[4]{13}\right)^2$$  

• $$3\sqrt[3]{2}$$  

Multiple Select

$$7^{\frac{3}{5}}$$ This fractional exponent can be written in TWO ways! Select two choices! 

• $$\left(\sqrt[3]{7}\right)^5$$  

• $$\left(\sqrt[5]{7}\right)^3$$  

• $$\sqrt[3]{7^5}$$  

• $$\sqrt[5]{7^3}$$  

Multiple Choice

Write the following as a radical expression: $$x^{\frac{3}{2}}$$  

(Be careful! ;)

• $$\sqrt[]{x^3}$$  

• $$\sqrt[2]{x^3}$$  

• $$\sqrt[3]{x^2}$$   

• $$\left(\sqrt[3]{x}\right)^2$$  

Multiple Choice

$$\sqrt[]{18}$$  

• $$3\sqrt[]{2}$$  

• $$2\sqrt[]{9}$$  

• $$2\sqrt[]{3}$$  

• $$\sqrt[]{9}$$  

• ​To simplify CUBE ROOTS:

  • Write the radicand as a product of numbers, one of which is a PERFECT CUBE.

  • Find the cube root of the perfect cube.​

Radical Expressions

Cube Root & Beyond