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}}