In this chapter we will learn about the concept of exponents with solved examples.
Exponents – What are they ?
Exponent is the way to represent repeated multiplication of number by itself.
The exponent is also called Power. So don’t get confused with the above two terms.
Suppose we multiply number 3 by itself 5 times.
\mathtt{\Longrightarrow \ 3\times 3\times 3\times 3\times 3}
Using exponents, this expression can be written as;
\mathtt{\Longrightarrow \ 3^{5}}
Hence, exponent is the short hand notation of multiplication expression in which numbers are multiplied by itself.
Base & Power in exponent function
In the exponent function, you will see large number in bottom and small number in the upper right corner.
The large number is called the Base.
The small number at the top is called Power / Exponent.
For example;
Consider the exponent function \mathtt{7^{3}}
Here number 7 is the base and number 3 is the exponent.
How to read exponents in words ?
There are many ways to read an exponent function.
Consider the expression \mathtt{9^{5}} .
You can read the exponent in following ways;
(i) “9 to the power 5”
(ii) ” 9 raised to the 5 “
(iii) ” 9 to the 5th “
(iv) ” 9 to the 5 “
Hence, there are multiple ways to express the exponent number. So don’t get confused with different sentences, they all mean the same thing.
How to simplify exponents ?
Just look at the base and power of given exponent function.
Here, base tells the number that is multiplied and power tells the number of times it is multiplied.
Let us understand this with examples;
Example 01
Exponent \mathtt{6^{5}}
Solution
Base = 6
It means number 6 is multiplied repeatedly.
Power = 5
It tells that the multiplication is done 5 times.
Simplifying the above exponents;
\mathtt{\Longrightarrow \ 6^{5}}\\\ \\ \mathtt{\Longrightarrow \ 6\times 6\times 6\times 6\times 6}\\\ \\ \mathtt{\Longrightarrow \ 7776}
Hence, 7776 is the solution of given exponents.
Example 02
Simplify exponent \mathtt{11^{3}}
Solution
Base is 11.
Here number 11 is multiplied repeatedly.
Exponent is 3.
It means that the number is multiplied thrice.
Simplifying the exponents;
\mathtt{\Longrightarrow \ 11^{3}}\\\ \\ \mathtt{\Longrightarrow \ 11\times 11\times 11}\\\ \\ \mathtt{\Longrightarrow \ 1331}
Hence, 1331 is the solution of given exponents.
Exponent function with negative base
There are two possibilities for exponents with negative base;
(a) Negative base with even power
In this case the final result is a positive number.
(b) Negative base with odd power
In this case the final result is negative number.
Both the above cases are concluded in below diagram.
Let us see some examples for further understanding.
Example 01
Find the value of exponent \mathtt{\ ( -2)^{3}}
Solution
Here the number has negative base and odd power. So the final result will be a negative number.
\mathtt{\Longrightarrow \ ( -2)^{3}}\\\ \\ \mathtt{\Longrightarrow \ -2\times -2\times -2}\\\ \\ \mathtt{\Longrightarrow \ -8}
Hence, -8 is the final solution.
Example 02
Find the value of exponent \mathtt{( -5)^{4}}
Solution
Here the number has negative base with even power. So the final result will be a positive number.
\mathtt{\Longrightarrow \ ( -5)^{4}}\\\ \\ \mathtt{\Longrightarrow \ -5\times -5\times -5\times -5}\\\ \\ \mathtt{\Longrightarrow \ 625}
Hence, 625 is the final solution.
Exponent function with negative power
The exponent with negative power can be converted into positive one by taking the reciprocal.
Let \mathtt{\ ( a)^{-n}} be the given exponent.
Then we can write;
\mathtt{\ ( a)^{-n} \ =\ \frac{1}{a^{n}}}
Let us see some examples related to this concept;
Example 01
Solve the exponent \mathtt{( 2)^{-3}} .
Solution
First convert the negative exponent into positive one by taking reciprocal.
\mathtt{\Longrightarrow \ ( 2)^{-3} \ }\\\ \\ \mathtt{\Longrightarrow \ \ \frac{1}{2^{3}}}
Now expand the exponent function.
\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 solution.
Example 02
Solve the exponent \mathtt{( 11)^{-2}}
Solution
Convert the power from negative into positive by taking reciprocal.
\mathtt{\Longrightarrow \ ( 11)^{-2} \ }\\\ \\ \mathtt{\Longrightarrow \ \ \frac{1}{11^{2}}}
Now calculate the value of exponent.
\mathtt{\Longrightarrow \ \ \frac{1}{11^{2}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1}{11\times 11}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1}{121}}
Hence, 1/121 is the solution.
I hope you understood all the above concepts. Let us now solve some problems.
Exponents – Solved Problems
(01) Find the value of below exponents.
\mathtt{( i) \ ( 2)^{5}}\\\ \\ \mathtt{( ii) \ ( -3)^{3}}\\\ \\ \mathtt{( iii) \ ( -5)^{2}}\\\ \\ \mathtt{( iv) \ ( 7)^{-4}}\\\ \\ \mathtt{( v) \ ( 10)^{0}}
Solution
\mathtt{( i) \ \ ( 2)^{5}}\\\ \\ \mathtt{\Longrightarrow 2\times 2\times 2\times 2\times 2\ }\\\ \\ \mathtt{\Longrightarrow \ 32}
\mathtt{( ii) \ ( -3)^{3}}\\\ \\ \mathtt{\Longrightarrow -3\times -3\times -3\ }\\\ \\ \mathtt{\Longrightarrow \ -27}
\mathtt{( iii) \ ( -5)^{2}}\\\ \\ \mathtt{\Longrightarrow -5\times -5}\\\ \\ \mathtt{\Longrightarrow \ 25}
\mathtt{( iv) \ ( 7)^{-4}}\\\ \\ \mathtt{\Longrightarrow \frac{1}{7^{4}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1}{7\mathtt{\times 7\times 7\times } 7}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1}{2401}}
\mathtt{( v) \ ( 10)^{0}}
Any number with power 0 results in number 1.
\mathtt{( 10)^{0} \ \Longrightarrow \ 1}
(02) Express the below numbers in exponent form
(i) -125
(ii) 64
(iii) 14641
(iv) 4/81
(v) -100/ 1728
Solution
(i) -125
\mathtt{\Longrightarrow \ -125}\\\ \\ \mathtt{\Longrightarrow \ -5\times -5\times -5}\\\ \\ \mathtt{\Longrightarrow \ ( -5)^{3}}
(ii) 64
\mathtt{\Longrightarrow \ 64}\\\ \\ \mathtt{\Longrightarrow \ 8\times 8}\\\ \\ \mathtt{\Longrightarrow \ ( 8)^{2}}
(iii) 14641
\mathtt{\Longrightarrow \ 14641}\\\ \\ \mathtt{\Longrightarrow \ 11\times 11\times 11\times 11}\\\ \\ \mathtt{\Longrightarrow \ ( 11)^{4}}
(iv) 4/81
\mathtt{\Longrightarrow \ \frac{4}{81}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2\times 2}{9\times 9}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2^{2}}{9^{2}}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{2}{9}\right)^{2}}
(v) -100/ 1728
\mathtt{\Longrightarrow \ \frac{-100}{1728}}\\\ \\ \mathtt{\Longrightarrow \ \frac{10\times 10}{-12\times -12\times -12}}\\\ \\ \mathtt{\Longrightarrow \ \frac{10^{2}}{-12^{3}}}