In this post we will learn to multiply exponents with different bases.

At the end of the chapter, solved examples are also provided for further clarity.

We have already covered the **multiplication of exponents with same base** in another chapter. Click the red link to read the same.

## Exponent basics

Before understanding the method to multiply exponents, let us first revise the basic concepts.

What are exponents ?

The exponent is used to **represent repeated multiplication of number by itself**.

For example, consider the below multiplication.

\mathtt{\Longrightarrow 3\times 3\times 3\times 3\times 3}

Note that here the number 3 is multiplied by itself five times.

Using exponents, the multiplication can be expressed as \mathtt{3^{5}}

Here the **large number 3 in between is called base**. It tells the number which is being multiplied.

The** small number at the top corner is called power/exponent**. It tells the number of time the number being multiplied.

## How to multiply exponent with different base ?

There are two cases in the given multiplication;

(a) the exponent have same power

(b) the exponent have different power

We will discuss both the cases in detail.

### Multiplication of exponent with different base but same power

The multiplication of exponent with different base and same power can be done by **multiplying the base separately and then inserting the same power**.

For example, consider the below multiplication;

\mathtt{\Longrightarrow \ a^{m} \times b^{m}}

Note that both the numbers have different base ” a ” & “b”, but have the same power “m”.

In this case, multiply the individual bases ” a ” & “b” and afterwards insert the power “m”.

\mathtt{a^{m} \times b^{m} \ =\ ( a\times b)^{m}}

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

**Example 01**

Multiply \mathtt{5^{3} \times 7^{3}} **Solution**

Note that both the numbers have same power.

To solve the expression, simply multiply the base and retain the given power.

\mathtt{\Longrightarrow \ 5^{3} \times 7^{3} \ \ }\\\ \\ \mathtt{\Longrightarrow \ ( 5\times 7)^{3}}\\\ \\ \mathtt{\Longrightarrow \ 35^{3}}

**Example 02**

Multiply \mathtt{-8^{11} \times 5^{11}} **Solution**

\mathtt{\Longrightarrow \ -8^{11} \times 5^{11} \ \ }\\\ \\ \mathtt{\Longrightarrow \ ( -8\times 5)^{11}}\\\ \\ \mathtt{\Longrightarrow \ -40^{11}}

**Example 03**

Multiply \mathtt{10^{-15} \times 6^{-15} \ }

**Solution**

\mathtt{\Longrightarrow \ 10^{-15} \times 6^{-15} \ \ }\\\ \\ \mathtt{\Longrightarrow \ ( 10\times 6)^{-15}}\\\ \\ \mathtt{\Longrightarrow \ 60^{-15}}

**Example 04**

Multiply \mathtt{a^{3} \times b^{3} \ }

**Solution**

\mathtt{\Longrightarrow \ a^{3} \times b^{3} \ \ }\\\ \\ \mathtt{\Longrightarrow \ ( a\times b)^{3}}\\\ \\ \mathtt{\Longrightarrow \ ( ab)^{3}}

### Multiplication of exponents with different base and power

The multiplication of exponent with different base and power is done by **first finding the individual value of exponent and then multiplying the numbers**.

Let us understand the concept with the help of example.**Example 01**

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

**Solution**

Note that both the multiplication have different base and power.

In this case, find value of exponent \mathtt{2^{3} \&\ 5^{2}} separately and then multiply.

\mathtt{\Longrightarrow \ 2^{3} \times \ 5^{2} \ \ }\\\ \\ \mathtt{\Longrightarrow \ 8\ \times \ 25}\\\ \\ \mathtt{\Longrightarrow \ 200\ }

Hence, **200 is the solution of given multiplication**.

**Example 02**

Multiply \mathtt{6^{-2} \times \ 3^{3}}

**Solution**

Both numbers have different base and power.

So we will first find the value of each exponent and then multiply.

\mathtt{\Longrightarrow \ 6^{-2} \times \ 3^{3} \ \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{1}{6^{2}} \ \times \ 27}\\\ \\ \mathtt{\Longrightarrow \ \frac{1}{36} \times 27}\\\ \\ \mathtt{\Longrightarrow \ \frac{27}{36}}

The number can be further simplified by dividing numerator and denominator by 9.

\mathtt{\Longrightarrow \ \frac{27}{36}}\\\ \\ \mathtt{\Longrightarrow \frac{27\div 9}{36\div 9}}\\\ \\ \mathtt{\Longrightarrow \ \frac{3}{4}}

Hence, **3/4 is the solution.**

**Example 03**

Multiply \mathtt{\ 2^{-2} \times \ 7^{-3}} **Solution**

\mathtt{\Longrightarrow \ 2^{-2} \times \ 7^{-3} \ \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{1}{2^{2}} \ \times \ \frac{1}{7^{3}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1}{4} \times \frac{1}{343}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1}{4\ \times 343}}\\\ \\ \mathtt{\Longrightarrow \frac{1}{1372}}