In this post we will learn about methods to multiply different exponents with solved examples.
This concept is widely used in algebra, so make sure you learn it properly.
Before understanding multiplication operation of exponents, let us revise the basic first.
Methods of Multiplying exponents
Let us understand the basics of exponents first.
When we multiply a number with itself we get an exponent.
Below are some examples:
\mathtt{2\ \times \ 2\ \times \ 2\ \times \ 2\ \times \ 2\ \ =\ 2^{5}}\\\ \\ \mathtt{a\ \times \ a\times \ a\ =\ a^{3}}\\\ \\ \mathtt{\frac{1}{3} \ \times \frac{1}{3} \ =\ \left(\frac{1}{3}\right)^{2}}
The number which is multiplied repeatedly is called Base.
Here numbers 2, a and 1/3 are the base.
Number of time a number is multiplied is shown by Power.
In \mathtt{\ 2^{5}} , the power is 5.
It signifies that number 2 is multiplied by itself 5 times.
I hope you understand the basics of exponents.
In this post we will discuss multiplication of following exponents:
(a) Integer exponents
(b) Exponents with fractions
(c) Exponents with negative power
Multiplying Integer Exponents
For integer exponents, three cases are possible:
(a) Integer exponents with same base and different power
(b) Different base and same power
(c) Both base and powers are different
Multiplying Integer exponents with same Base
When we multiply exponent with same base, we simply add the power of given exponents.
Remember the Formula;
\mathtt{a^{m} \times \ a^{n} \ =\ a^{m\ +n}}
Validating the Formula
Multiply \mathtt{2^{3} \times \ 2^{4}}\\\ \\
\mathtt{2^{3} =\ 2\ \times 2\times 2\ =\ 8}\\ \\ \mathtt{2^{4} =\ 2\ \times 2\times 2\times 2\ =\ 16}\\\ \\ \mathtt{2^{3} \times \ 2^{4} \ \Longrightarrow \ 8\ \times 16\ \Longrightarrow \ 128}
Now use the above formula
\mathtt{2^{3} \times \ 2^{4} \ =\ 2^{3\ +4} =2^{7} =128}
Hence, the formula is proved.
Example 01
Multiply \mathtt{9^{3} \times 9^{4} \times 9^{2}}
Solution
\mathtt{\Longrightarrow \ 9^{3} \times 9^{4} \times 9^{2}}\\\ \\ \mathtt{\Longrightarrow \ 9^{3+4+2}}\\\ \\ \mathtt{\Longrightarrow \ 9^{9} \ }
Example 02
Multiply \mathtt{\Longrightarrow \ 5^{2} \times \ 5^{5} \times \frac{1}{( 5)^{3}}}
Solution
\mathtt{\frac{1}{( 5)^{3}}} can be written as \mathtt{5^{-3}}
Now multiplying the exponents
\mathtt{\Longrightarrow \ 5^{2} \times \ 5^{5} \times \ 5^{-3}}\\\ \\ \mathtt{\Longrightarrow \ 5^{2+5-3}}\\\ \\ \mathtt{\Longrightarrow \ 5^{4} \ }
Example 03
Multiply \mathtt{7 \times \ 7^{4} \times \ 1}
Solution
1 can be written as 7 ^ {0}
\mathtt{\Longrightarrow \ 7^{1} \times \ 7^{4} \times \ 7^{0}}\\\ \\ \mathtt{\Longrightarrow \ 7^{1+4+0}}\\\ \\ \mathtt{\Longrightarrow \ 7^{5}}
Different base with same power
The multiplication of exponents with different base and same power can done by multiplying the base together keeping the powers same.
For Example, Let \mathtt{a^{m} \ \&\ b^{m}} are the exponents with different base and same power m.
The multiplication of exponents is given as:
Remember the formula;
\mathtt{a^{m} \ \times \ b^{m} \ =\ ( a\times b)^{m\ }}
Let us look at some of the examples.
Example 01
Multiply \mathtt{3^{5} \times 6^{5}}
Solution
\mathtt{\Longrightarrow 3^{5} \times 6^{5}}\\\ \\ \mathtt{\Longrightarrow \ ( 3\ \times \ 6)^{5}}\\\ \\ \mathtt{\Longrightarrow \ 18^{5}}
Example 02
Multiply \mathtt{2^{6} \times 3^{6} \times 9^{6}}
Solution
\mathtt{\Longrightarrow 2^{6} \times 3^{6} \times 9^{6}}\\\ \\ \mathtt{\Longrightarrow \ ( 2\ \times \ 3\times 9)^{6}}\\\ \\ \mathtt{\Longrightarrow \ 54^{6}}
Example 03
\mathtt{11^{12} \times 10^{12} \times 3^{12}}
Solution
\mathtt{\Longrightarrow 11^{12} \times 10^{12} \times 3^{12}}\\\ \\ \mathtt{\Longrightarrow \ ( 11\ \times \ 10\times 3)^{12}}\\\ \\ \mathtt{\Longrightarrow \ 330^{12}}
Different base with different Powers
When we have exponents with different base and power, we have to simplify the individual exponents in normal digits and then multiply the numbers.
There is no specific formula for this case.
We simply have to apply rudimentary technique of basic multiplication.
Example 01
Multiply \mathtt{2^{3} \times 4^{2}}
Solution
Here both the exponents have different bases and powers, so we do basic multiplication.
Follow the below steps;
(i) Find individual exponent value
\mathtt{2^{3} \Longrightarrow \ 2\ \times 2\times 2\Longrightarrow \ 8}\\\ \\ \mathtt{4^{2} \Longrightarrow \ 4\ \times \ 4\ \Longrightarrow \ 16}
(ii) Multiply the values
\mathtt{\Longrightarrow 2^{3} \times 4^{2}}\\\ \\ \mathtt{\Longrightarrow \ 8\ \times 16}\\\ \\ \mathtt{\Longrightarrow \ 128}
Example 02
Multiply \mathtt{\Longrightarrow 2^{4} \times 3^{2} \times 5^{3}}
Solution
Al the exponents have different base and powers
(i) Find individual value of exponents
\mathtt{2^{4} \Longrightarrow 2\times 2\times 2\times 2\Longrightarrow 16}\\\ \\ \mathtt{3^{2} \Longrightarrow 3\times 3\Longrightarrow \ 9}\\\ \\ \mathtt{5^{3} \Longrightarrow 5\times 5\times 5\Longrightarrow 125}
(ii) Multiply the values
\mathtt{\Longrightarrow 2^{4} \times 3^{2} \times 5^{3}}\\\ \\ \mathtt{\Longrightarrow \ 16\ \times \ 9\ \times 125}\\\ \\ \mathtt{\Longrightarrow \ 18000}
Multiplying exponents with fractions
There are three similar cases for fraction exponents;
(a) Exponents with same base
(b) Exponent with different base with same power
(c) Exponents with different base with different power
All the cases are solved in same manner as the normal exponents.
Due to fraction there are slight changes which is discussed below.
Exponent with same base
When the fractional bases are same, the multiplication is done simply by adding the power.
Example 01
Multiply \mathtt{\left(\frac{1}{3}\right)^{4} \times \left(\frac{1}{3}\right)^{3}}
Solution
\mathtt{\Longrightarrow \ \left(\frac{1}{3}\right)^{4+3}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{1}{3}\right)^{7}}
Example 02
\mathtt{\left(\frac{6}{11}\right)^{2} \times \left(\frac{6}{11}\right)^{9} \ }
Solution
\mathtt{\Longrightarrow \ \left(\frac{6}{11}\right)^{2+9}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{6}{11}\right)^{11}}
Different fraction base but same power
In this case, both the numerators and denominators are multiplied and the power is kept the same.
Example 01
\mathtt{\left(\frac{2}{3}\right)^{2} \times \left(\frac{4}{5}\right)^{2} \ }
Solution
\mathtt{\Longrightarrow \ \left(\frac{2\times 4}{3\times 5}\right)^{2}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{8}{15}\right)}
Exponents with different fractional base and power
In this case we have to simplify the individual exponents and then multiply together.
Example
\mathtt{\left(\frac{1}{3}\right)^{3} \times \left(\frac{5}{6}\right)^{2}}
Solution
\mathtt{\left(\frac{1}{3}\right)^{3} \Longrightarrow \ \frac{1}{3} \ \times \frac{1}{3} \times \frac{1}{3} \ \Longrightarrow \ \frac{1}{27}}\\\ \\ \mathtt{\left(\frac{5}{6}\right)^{2} \ \Longrightarrow \frac{5}{6} \times \frac{5}{6} \Longrightarrow \ \frac{25}{36}}\\\ \\ \mathtt{\left(\frac{1}{3}\right)^{3} \times \left(\frac{5}{6}\right)^{2} \ \Longrightarrow \frac{1}{27} \times \frac{25}{36}}\\\ \\ \mathtt{\left(\frac{1}{3}\right)^{3} \times \left(\frac{5}{6}\right)^{2} \ \Longrightarrow \frac{25}{972}}
Multiplying exponents with negative power
Below cases possible for negative powers
(a) Base are same
(b) Base are different but power is same
(c) Both base and Power are different
Multiplying negative exponents with same base
In this case, simply add the powers of the given exponents.
Example 01
Multiply the exponents; \mathtt{( 5)^{-3} \times ( 5)^{-2} \ }
Solution
\mathtt{\Longrightarrow \ ( 5)^{( -3) \ +\ ( -2)}}\\\ \\ \mathtt{\Longrightarrow \ 5^{-3-2}}\\\ \\ \mathtt{\Longrightarrow \ 5^{-5}}
Example 02
\mathtt{( 10)^{-7} \times ( 10)^{-4}}
Solution
\mathtt{\Longrightarrow \ ( 10)^{( -7) \ +\ ( -4)}}\\\ \\ \mathtt{\Longrightarrow \ 10^{-7-4}}\\\ \\ \mathtt{\Longrightarrow \ 10^{-11} \ }
Multiplying negative exponents with different base but same power
In this case simply multiply the given bases and keep the exponents as it is.
Example 01
\mathtt{( 11)^{-3} \times ( 12)^{-3}}
Solution
\mathtt{\Longrightarrow \ ( 11\times 12)^{( -3) \ }}\\\ \\ \mathtt{\Longrightarrow \ 132^{-3}}
Example 02
\mathtt{( 5)^{-4} \times ( 7)^{-4} \times \ ( 2)^{-4}}
Solution
\mathtt{\Longrightarrow \ ( 5\times 7\times 2)^{( -4) \ }}\\\ \\ \mathtt{\Longrightarrow \ 70^{-4}}
Multiplying exponents with different base and power
In this case, you have to calculate individual value of the exponents and then multiply.
Example
\mathtt{( 3)^{-2} \times ( 2)^{-3}}
Solution
\mathtt{( 3)^{-2} \ \Longrightarrow \ \frac{1}{3^{2}} \ \Longrightarrow \ \frac{1}{9}}\\\ \\ \mathtt{( 2)^{-3} \ \Longrightarrow \ \frac{1}{2^{3}} \ \Longrightarrow \ \frac{1}{8}}\\\ \\ \mathtt{( 3)^{-2} \times ( 2)^{-3} \ \Longrightarrow \ \frac{1}{9} \times \frac{1}{8}}\\\ \\ \mathtt{( 3)^{-2} \times ( 2)^{-3} \ \Longrightarrow \ \frac{1}{72}}