# Exponent rules

Here you will learn about exponent rules, including what each exponent rule is and how to apply an exponent rule to an exponential expression.

Students will first learn about exponent rules as part of expressions and equations in grade 8.

## What are exponent rules?

Exponent rules are mathematical laws that help us simplify expressions involving powers or exponents. Exponent rules are also referred to as “laws of exponents” or “properties of exponents.”

There are many different rules of exponents; these involve properties such as multiplication, division, raising to a power, and dealing with zero and negative exponents. These rules help us work with exponents in a more organized way, making math easier and letting us solve all sorts of problems efficiently.

## Common Core State Standards

How does this relate to 8 th grade math?

• Grade 8 – Expressions and Equations (8.EE.A.1)
Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 \times 3-5=3-3= \cfrac{1}{33}=\cfrac{1}{27}.

## How to use exponent rules

In order to simplify an expression using exponent rules:

1. Identify the base and the exponent.
2. Apply the appropriate exponent rule(s) according to the operation.
3. Perform the arithmetic operations indicated by the exponent rules to simplify the expression.

## Exponent rules examples

### Example 1: product rule

Evaluate the expression.

3^3 \times 3^2

1. Identify the base and the exponent.

Base (first term): 3

Base (second term): 3

Exponent: (first term): 3

Exponent: (second term): 2

2Apply the appropriate exponent rule(s) according to the operation.

Since you are multiplying two terms with the same base, you can use the product rule which says to add the two exponents: a^m \times a^n=a^{m+n}

3Perform the arithmetic operations indicated by the exponent rules to simplify the expression.

\begin{aligned}&3^3 \times 3^2=3^{3+2}=3^5 \\\\ &3^5=3 \times 3 \times 3 \times 3 \times 3=243\end{aligned}

### Example 2: quotient rule

Evaluate the expression.

6^5 \div 6^3

Identify the base and the exponent.

Apply the appropriate exponent rule(s) according to the operation.

Perform the arithmetic operations indicated by the exponent rules to simplify the expression.

### Example 3: power of a power rule

Evaluate the expression.

\left(10^2\right)^3

Identify the base and the exponent.

Apply the appropriate exponent rule(s) according to the operation.

Perform the arithmetic operations indicated by the exponent rules to simplify the expression.

### Example 4: power of a product rule

Evaluate the expression.

(4 \times 3)^2

Identify the base and the exponent.

Apply the appropriate exponent rule(s) according to the operation.

Perform the arithmetic operations indicated by the exponent rules to simplify the expression.

### Example 5: power of a quotient rule

Evaluate the expression.

(12 \div 4)^6

Identify the base and the exponent.

Apply the appropriate exponent rule(s) according to the operation.

Perform the arithmetic operations indicated by the exponent rules to simplify the expression.

### Example 6: zero exponent rule

Evaluate the expression.

18^0

Identify the base and the exponent.

Apply the appropriate exponent rule(s) according to the operation.

Perform the arithmetic operations indicated by the exponent rules to simplify the expression.

### Example 7: negative exponent rule

Evaluate the expression.

5^{-4}

Identify the base and the exponent.

Apply the appropriate exponent rule(s) according to the operation.

Perform the arithmetic operations indicated by the exponent rules to simplify the expression.

### Example 8: fractional exponent rule

Evaluate the expression.

9^{3 / 2}

Identify the base and the exponent.

Apply the appropriate exponent rule(s) according to the operation.

Perform the arithmetic operations indicated by the exponent rules to simplify the expression.

### Teaching tips for exponent rules

• Begin by ensuring students understand what exponents represent. Reinforce the idea that exponents indicate repeated multiplication.

• Start with small numbers (bases and exponents) and gradually progress to larger numbers and more complex expressions.

• Print out an anchor chart listing each exponent rule for students to refer to when completing worksheets.

### Easy mistakes to make

• Ignoring the order of operations
When simplifying exponential expressions, it is important to keep note of the order of operations, especially keeping an eye out for parentheses in which anything inside should be performed first.

• Forgetting the zero exponent rule
Students commonly forget that a non-zero base raised to the zero power equals 1, which can lead to calculation errors.

• Confusing product and power rules.
Confusing the product rule (adding exponents when multiplying powers with the same base) with the power rule (multiplying exponents when raising a power to another power).

### Practice exponent rules questions

1. Which expression is equivalent to   (4 \times 8)^2 ?

4^2+8^2

4^2 \times 8^2

32+32

(4 \times 8)+(4 \times 8)

The power of a product rule states that when you have a product of multiple numbers raised to a power, you can distribute that power to each number in the product.

Therefore, (4 \times 8)^2=4^2 \times 8^2

2. Solve the equation.

15+12^0= \; ?

15

27

16

0

The zero rule says that any number raised to the power of zero equals one. Therefore, 12^0=1

When you add 1 to 15, you get 16.

3. Which expression is equivalent to 5^2 \times 5^9?

5^{11}

5^{18}

10 \times 45

5^7

The product rule says that when multiplying two numbers with the same base, add their exponents.

Therefore, 5^2 \times 5^9=5^{(2+9)}=5^{11}

4. Which expression is equivalent to (9 \div 3)^3?

6^3

(9 \div 3)+(9 \div 3)+(9 \div 3)

9^3-3^3

9^3 \div 3^3

The power of a quotient rule states that when you have a quotient (division) of two numbers raised to a power, you can distribute that power to both the numerator and the denominator (the dividend and the divisor).

That means that since the entire division expression of 9 \div 3 is being cubed, or raised to the third power, we can distribute the power to the 9 and the 3.

Then we can solve:

(9 \div 3)^3=9^3 \div 3^3=729 \div 27=27

5. Which exponent is equivalent to \left(6^3\right)^5?

6^8

6^2

6^{15}

6^{35}

The power of a power rule states that when raising a power to another power, you multiply the exponents.

Therefore, \left(6^3\right)^5=6^{3 \times 5}=6^{15}

6. How would you use the quotient rule to solve 8^9 \div 8^7?

Add the exponents to make it 8^{16}.

Subtract the exponents to make it 8^{2}.

Turn the exponents into a fraction to make it 8^{\frac{9}{7}}.

Turn the exponents into a negative exponent to make it 8^{-2}.

The quotient rule states that when dividing two numbers with the same base, subtract their exponents.

Therefore, 8^9 \div 8^7=8^{9-7}=8^2

7. Which is equivalent to 10^{-5}?

–50

–100,000

-10^5

\cfrac{1}{10^5}

The negative exponent rule states that a number raised to a negative exponent is the reciprocal of that number raised to the positive exponent.

Therefore, 10^{-5}=\cfrac{1}{10^5}

8. Which is equivalent to 6^{\frac{4}{5}}?

\sqrt[5]{6^4}

–\sqrt[4]{6^5}

-6 \times \cfrac{4}{5}

6^{20}

The fractional exponent rule states that when you have a number raised to a fractional exponent, you take the root indicated by the denominator and raise the base to the power indicated by the numerator.

Therefore, 6^{\frac{4}{5}}=\sqrt[5]{6^4}

## Exponent rules FAQs

What are the rules of exponents?

Exponent rules are mathematical laws that help us simplify expressions involving powers or exponents. Exponent rules are also referred to as “laws of exponents” or “properties of exponents.”

Which exponent rules involve negative numbers?

The negative exponent rule states that any number raised to a negative power can be rewritten as the reciprocal of the same number raised to the corresponding positive power.

Do exponent rules only apply to real numbers?

No, exponent rules apply to various types of numbers, including real numbers, complex numbers, rational numbers, and more.

How are exponent rules related to logarithms?

Exponent rules and logarithms are closely related because they are inverse operations of each other. Logarithms allow us to solve for exponents, and exponent rules help simplify expressions involving logarithms.

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