## Table of Contents

- How Do You Multiply Exponents?
- Multiplying Exponents with the Same Base: Do You Add Exponents When Multiplying?
- Multiplying Exponents With Different Bases
- Multiplying Exponents Rules: All Exponent Rules
- Lesson Summary

Learn about multiplying exponents with different bases, as well as multiplying exponents with the same base. Explore multiplying exponents rules, how to multiply exponents with different bases, and what happens when you multiply exponents.
Updated: 11/08/2021

- How Do You Multiply Exponents?
- Multiplying Exponents with the Same Base: Do You Add Exponents When Multiplying?
- Multiplying Exponents With Different Bases
- Multiplying Exponents Rules: All Exponent Rules
- Lesson Summary

This lesson will focus on operations on numbers that have exponents. In particular, the main goal of this lesson will be to answer the question of when and how do you multiply exponents. After establishing the when and how there will be an exploration of multiplying exponents rules that outline how to combine exponents in a variety of situations.

The four basic operations of arithmetic are addition, subtraction, multiplication, and division. Recall that multiplication is simply a shorter way to write repeated addition. For example, {eq}3 + 3 + 3 + 3 + 3 + 3 = 6 \times 3 {/eq}. Similarly, an **exponent** is an operation that is a shorter way to write repeated multiplication. For example, {eq}3 \times 3 \times 3 \times 3 \times 3 \times 3 = 3^6 {/eq}. In this case, 6 is called the exponent and 3 is called the **base**. At times, the word **power** is also used to describe an exponent. For instance, the expression in Fig. 1 might be called "two to the power of five."

More generally, any exponential expression can be written as {eq}b^a {/eq} where *a* is the exponent and *b* is the base.

What do you do when multiplying exponents? Since exponents mean repeated multiplication, a natural question of what happens when you multiply exponents might arise. In this section, the idea of multiplying two bases raised to exponents will be explored. For example, consider the expression {eq}2^5 \times 2^3 {/eq}. It may be tempting here to assume that the 2's multiply together or that the 5 and 3 multiply together, but in fact, neither of these is correct. Instead, lean on the definition of an exponent to rewrite this expression before combining any terms:

{eq}2^5 \times 2^3 = (2 \times 2 \times 2 \times 2 \times 2) \times (2 \times 2 \times 2) {/eq}

Noticing that there are eight total 2's here, this expression can be rewritten as {eq}2^8 {/eq}. This idea can be generalized to show that you add exponents when multiplying like bases: {eq}b^x \times b^y = b^{x + y} {/eq}

But what happens when multiplying exponents with different bases? This cannot be done by adding the exponents as in the previous section because it is unclear what to do with the base. This section will show how to multiply exponents with different bases using two different methods of simplification.

The first method is the simplification method. Essentially, this method is used to write out the exponential expression as a long string of multiplication and then combine all of the terms. Consider the example: {eq}3^4 \times 4^3 {/eq}. Writing this out gives: {eq}3 \times 3 \times 3 \times 3 \times 4 \times 4 \times 4 {/eq} and none of the 3's can be combined with the 4's other than by multiplying out this product. Doing so gives an answer of 5184.

The second method of multiplying exponential expressions with different bases is grouping. The grouping method essentially says to rearrange the terms so that the multiplication can be done a bit more easily. Recall the example above of {eq}3^4 \times 4^3 {/eq} Notice that there is one more 3 than there are 4's. So we can combine the same number of 3's and 4's and have one 3 left over. For instance:

{eq}3 \times 3 \times 3 \times 3 \times 4 \times 4 \times 4 = 3 \times 4 \times 3 \times 4 \times 3 \times 4 \times 3 = 12 \times 12 \times 12 \times 3 = 12^3 \times 3 {/eq}

So it is possible to rewrite the expression using exponents: {eq}3^4 \times 4^3 = 3^3 \times 4^3 \times 3 = 12^3 \times 3 {/eq}.

More generally, it can be said that {eq}a^x \times b^x = (ab)^x {/eq} or, in words, multiplying different bases raised to the same power means that the bases can be multiplied together first and then raised to that power.

Notice, though, that this grouping method requires having the same exponent on each base. For instance, to group {eq}x^5 \times y^7 {/eq}, first rewrite it as {eq}x^5 \times y^5 \times y^2 {/eq} and then group the terms with the same exponents: {eq}(xy)^5 \times y^2 {/eq}

There are seven rules to remember for multiplying bases with exponents. Each multiplying exponent rule will be shown below with an example to help illustrate it. Some of these rules do not involve multiplication but will be useful in solving more complicated problems involving multiplying exponents.

If multiplying exponential expressions with the same base, then add the exponents together: {eq}2^4 \times 2^7 = 2^{11} {/eq}

In general: {eq}b^x \times b^y = b^{x + y} {/eq}

If multiplying exponential expressions with the same base leads to adding the powers together, then it makes sense that dividing exponential expressions with the same base leads to subtracting the powers: {eq}2^7 \div 2^5 = 2^2 {/eq}

This can readily be seen by writing out the exponents the long way and canceling like terms in a fraction: {eq}2^7 \div 2^5 = \frac{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2}{2 \times 2 \times 2 \times 2 \times 2} = 2 \times 2 = 2^2 {/eq}

In general: {eq}b^x \div b^y = b^{x - y} {/eq}

If an exponential expression is raised to another power, then multiply the powers together. For example: {eq}(2^3)^4 = 2^{12} {/eq}

This can be seen by writing out the long-form multiplication and then using rule 1: {eq}(2^3)^4 = 2^3 \times 2^3 \times 2^3 \times 2^3 = 2^{3 + 3 + 3 + 3} = 2^{12} {/eq}

In general: {eq}(b^x)^y = b^{xy} {/eq}

If a product is raised to a power, then each factor of the product can be raised to that power individually. For example, {eq}(4 \times 3)^2 = 4^2 \times 3^2 {/eq}

Again, this can be seen by writing out the long-form multiplication and then grouping the same bases: {eq}(4 \times 3)^2 = (4 \times 3) \times (4 \times 3) = 4 \times 4 \times 3 \times 3 = 4^2 \times 3^2 {/eq}

In general: {eq}(ab)^x = a^x \times b^x {/eq}

Similar to Rule 4, if a quotient is raised to a power, then the numerator and denominator can be raised to that power separately. For example: {eq}(\large \frac{2}{3})^4 = \frac{2^4}{3^4} {/eq}

In general: {eq}(\large \frac{a}{b})^x = \frac{a^x}{b^x} {/eq}

Any nonzero term raised to the power of zero is equal to 1. For example: {eq}2^0 = 1 {/eq} or {eq}(xy)^0 = 1 {/eq}.

This rule is a consequence of Rule 2 above: {eq}2^3 \div 2^3 = 2^{3 - 3} = 2^0 {/eq}, but it is also known that dividing an expression by itself should yield an answer of 1.

In general: {eq}b^0 = 1 {/eq}

A number raised to a negative power equals the reciprocal of that number raised to a positive power. For example: {eq}3^{-4} = \frac{1}{3^4} {/eq}

This rule is also a consequence of Rule 2.

In general: {eq}b^{-n} = \frac{1}{b^n} {/eq}

In Fig 2 we can see all of the exponent rules in one place.

An **exponent** (also called a **power**) is a symbol used to denote repeated multiplication. For example, {eq}3^7 {/eq} means to multiply 7 copies of the number 3. In this case, the 7 is called the exponent. The **base** of an exponential expression is the number being multiplied repeatedly. In this case, the 3 is the base.

Since exponents represent repeated multiplication, exponents have special interactions when exponential expressions are multiplied together. To multiply exponential expressions with like bases, simply add the exponents. To multiply exponential expressions with unlike bases, either evaluate each independently or combine some of the terms by grouping. There are serval other exponent rules that can help in evaluating exponential expressions and the chart in Fig. 2 outlines each one.

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