# Product rule for exponents

In this chapter we will learn about product of power rule of exponents in detail with solved examples.

To understand the rule, you should have basic understanding of the concept of exponents. Click the red link to read about exponents in detail.

## Exponent Product rule

According to the rule, the multiplication of exponent with same base can be done by adding the exponents by keeping the original base.

Consider the multiplication of numbers \mathtt{a^{m} \times a^{n}} .

Note that both the numbers have same base ” a “.

The multiplication is given as;

\mathtt{a^{m} \times a^{n} =a^{m+n}}

Here we have simply added the exponents to get the solution.

### Can we apply Product of power rule when the numbers have different base ?

No !!!

This rule is only applicable for exponents with same base.

For example, consider the multiplication \mathtt{a^{m} \times \ b^{n\ }} .

Note that both numbers have different base. So we can’t apply the product rule of exponents.

### Verification of Product law of exponents

Consider the multiplication of number \mathtt{2^{3} \times 2^{2}}

Let’s first solve the expression with conventional method.

\mathtt{\Longrightarrow \ 2^{3} \times 2^{2}}\\\ \\ \mathtt{\Longrightarrow \ ( 2\times 2\times 2) \times ( 2\times 2)}\\\ \\ \mathtt{\Longrightarrow \ 8\ \times 4}\\\ \\ \mathtt{\Longrightarrow \ 32}

Hence, 32 is the solution of given expression.

Now let’s solve the same expression using Exponent product rule.

\mathtt{\Longrightarrow \ 2^{3} \times 2^{2}}\\\ \\ \mathtt{\Longrightarrow \ 2^{3\ +\ 2}}\\\ \\ \mathtt{\Longrightarrow \ 2^{5}}\\\ \\ \mathtt{\Longrightarrow \ 32}

Note that we get the same solution as the conventional method.

Hence, the law is verified.

### Case of exponent power rules

Given below are some cases which we will encounter while solving different problems.

In all the below cases, the process of solving is same as discussed above.

(a) Base is fraction

In the multiplication, we will keep the base same and subtract the exponents.

\mathtt{\ \left(\frac{a}{b}\right)^{m} \times \left(\frac{a}{b}\right)^{n} =\ \left(\frac{a}{b}\right)^{m+n}}

(b) Base is negative number

\mathtt{\ ( -a)^{m} \times ( -a)^{n} =\ ( -a)^{m+n}}

(c) Exponent is negative number

\mathtt{\Longrightarrow \ ( a)^{m} \times ( a)^{-n}}\\\ \\ \mathtt{\Longrightarrow \ ( a)^{m+\ ( -n)}}\\\ \\ \mathtt{\Longrightarrow ( a)^{m-n}}

I hope you understood the above concept. Let us solve some problems for further clarity.

## Exponent product rule – Solved Examples

Example 01
Multiply \mathtt{5^{3} \times 5^{2}}

Solution
Note that both the numbers have same base ” 5 “.

To multiply the numbers, you simply have to add the exponents.

\mathtt{\Longrightarrow \ 5^{3} \times 5^{2}}\\\ \\ \mathtt{\Longrightarrow \ 5^{3\ +\ 2}}\\\ \\ \mathtt{\Longrightarrow \ 5^{5}}

Example 02
Multiply \mathtt{( -6)^{4} \times ( -6)^{5}}

Solution
Note that both the numbers have same base.

Applying product of exponent rule.

\mathtt{\Longrightarrow \ ( -6)^{4} \times ( -6)^{5}}\\\ \\ \mathtt{\Longrightarrow \ ( -6)^{4\ +\ 5}}\\\ \\ \mathtt{\Longrightarrow \ ( -6)^{9}}

Example 03
Multiply \mathtt{\ \left(\frac{5}{3}\right)^{10} \times \left(\frac{5}{3}\right)^{17}}

Solution
Both the numbers have same fractional number base.

\mathtt{\Longrightarrow \ \left(\frac{5}{3}\right)^{10} \times \left(\frac{5}{3}\right)^{17}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{5}{3}\right)^{10\ +\ 17}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{5}{3}\right)^{27}}

Example 04
Multiply \mathtt{\left( x^{2} yz\right)^{51} \times \left( x^{2} yz\right)^{25}}

Solution
The base are in the form of variables.

We will apply the product rule as both the number have same base.

\mathtt{\Longrightarrow \ \left( x^{2} yz\right)^{51} \times \left( x^{2} yz\right)^{25}}\\\ \\ \mathtt{\Longrightarrow \ \left( x^{2} yz\right)^{51+\ 25}}\\\ \\ \mathtt{\Longrightarrow \ \left( x^{2} yz\right)^{76}}

Example 05
Multiply \mathtt{( 2)^{-11} \times ( 2)^{-23}}

Solution
\mathtt{\Longrightarrow \ ( 2)^{-11} \times ( 2)^{-23}}\\\ \\ \mathtt{\Longrightarrow \ ( 2)^{-11+\ ( -23)}}\\\ \\ \mathtt{\Longrightarrow \ ( 2)^{-11-23}}\\\ \\ \mathtt{\Longrightarrow \ ( 2)^{-34}}

Example 06
Multiply \mathtt{\left(\frac{-10}{3}\right)^{-5} \times \left(\frac{-10}{3}\right)^{7}}

Solution
\mathtt{\Longrightarrow \ \left(\frac{-10}{3}\right)^{-5} \times \left(\frac{-10}{3}\right)^{7}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{-10}{3}\right)^{-5+\ 7}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{-10}{3}\right)^{2}}

Example 07
Multiply \mathtt{( 8)^{91} \times ( 8)^{-63}}

Solution
\mathtt{\Longrightarrow \ ( 8)^{91} \times ( 8)^{-63}}\\\ \\ \mathtt{\Longrightarrow \ ( 8)^{91-63}}\\\ \\ \mathtt{\Longrightarrow \ ( 8)^{28}}