# Multiplying exponent with different bases

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.

## 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}}