Negative exponent rules

In this chapter we will learn about negative exponents and its calculation method with solved examples.

Exponent basics

We know that a exponent function contain two numbers, base and exponent.

The base tells the number which is multiplied repeatedly.

The exponent/power tells the number of time the base is multiplied.

What are negative exponents ?

The exponent function in which the power is a negative number is called negative exponent.

Some examples of negative exponent are given below;

\mathtt{\Longrightarrow ( 8)^{-2}}\\\ \\ \mathtt{\Longrightarrow \ ( 10)^{-15}}\\\ \\ \mathtt{\Longrightarrow \ ( 3)^{-9}}

Can a fraction number have negative exponents ?

Yes !!!

Any possible number such as fractions, decimals, integers etc. can have negative powers and are called negative exponents.

Some examples of negative exponents with fraction base are;

\mathtt{\Longrightarrow \left(\frac{3}{4}\right)^{-5}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{1}{7}\right)^{-51}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{16}{5}\right)^{-99}}

How to convert negative exponent into positive one ?

Negative exponent can be converted into positive exponent by taking reciprocal of the base.

Let us understand this with example.

Consider the negative exponent \mathtt{\ 5^{-11}} .

To convert into positive exponent, take the reciprocal of base 5 and change the negative power into positive one.

\mathtt{\Longrightarrow \ 5^{-11}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1}{5^{11}}}

How to convert the negative exponent with fraction base into positive one ?

The process is same as above. We have to take reciprocal of the numbers.

During this process, the numerator & denominator digits will be switched.

For example, consider the fractional negative exponent \mathtt{\left(\frac{5}{3}\right)^{-7}} .

To convert the number into positive exponent, swap the position of numerator and denominator.

\mathtt{\Longrightarrow \ \left(\frac{5}{3}\right)^{-7}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{3}{5}\right)^{7}}

Example 02
Convert into positive exponent \mathtt{\frac{1}{7^{-5}}}

Solution
Take the reciprocal of base 7.

\mathtt{\Longrightarrow \ \frac{1}{7^{-5}}}\\\ \\ \mathtt{\Longrightarrow ( 7)^{5}}

Finding the value of negative exponents

Follow the below steps to find the value of negative exponent;

(a) Convert negative exponent into positive one by taking reciprocal.

(b) Now multiply the base repeatedly as stated by the power value to get the solution.

I hope you understood the above process. Let us see some examples for further clarity.

Example 01
Find value of \mathtt{2^{-3}}

Solution

\mathtt{\Longrightarrow \ \frac{1}{2^{3}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1}{2\times 2\times 2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1}{8}}

Hence, 1/8 is the value of given exponent.

Example 2
Find the value of \mathtt{\frac{1}{12^{-2}}}

Solution
\mathtt{\Longrightarrow \ \frac{1}{12^{-2}}}\\\ \\ \mathtt{\Longrightarrow 12^{2}}\\\ \\ \mathtt{\Longrightarrow \ 12\ \times \ 12}\\\ \\ \mathtt{\Longrightarrow \ 144}

Hence, 144 is the value of given negative exponent.

Example 03
Find the value of \mathtt{\left(\frac{4}{7}\right)^{-3}}

Solution
First convert the given number into positive exponent.

\mathtt{\Longrightarrow \ \left(\frac{4}{7}\right)^{-3}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{7}{4}\right)}

Now find the values of the exponents.

\mathtt{\Longrightarrow \ \left(\frac{7}{4}\right)^{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{7\times 7\times 7}{4\times 4\times 4}}\\\ \\ \mathtt{\Longrightarrow \ \frac{343}{64}}

Hence, 343/64 is the value.

Example 04
Find the value of \mathtt{( -11)^{-2}}

Solution

\mathtt{\Longrightarrow \ \frac{1}{( -11)^{2}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1}{-11\times -11}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1}{121}}

Example 05
Find value of \mathtt{\left(\frac{-3}{13}\right)^{-3}}

Solution

\mathtt{\Longrightarrow \ \left(\frac{-3}{13}\right)^{-3}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{13}{-3}\right)^{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{13\times 13\times 13}{-3\times -3\times -3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2197}{-27}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-2197}{27}}

Example 06
Find value of \mathtt{\left(\frac{-2}{5}\right)^{-4}}

Solution

\mathtt{\Longrightarrow \ \left(\frac{-2}{5}\right)^{-4}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{5}{-2}\right)^{4}}\\\ \\ \mathtt{\Longrightarrow \ \frac{5\times 5\times 5\times 5}{-2\times -2\times -2\times -2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{625}{16}}