In this chapter we will learn about different rules of exponent with solved examples.
These laws are important as they are required to solve complex algebraic problems in higher classes. So make sure you learn this chapter very well.
To understand this chapter, you should first have basic understanding of exponents. Click the red link to read the basics.
Rules of Exponents
Here we will discuss following laws of exponents.
(a) Product of power rule
(b) Quotient of power rule
(c) Power of Power rule
(d) Powers of Product rule
(e) Powers of Quotient rule
(f) Zero power rule
(g) Negative exponent rule
Product of Power rule for exponents
When two or more exponent numbers with same base are multiplied then we get the final result by keeping the same base and adding the exponents.
In short we can say that, multiplication of exponents with same base can be done by adding the powers.
Let \mathtt{a^{b} \ \&\ a^{c}} are the two numbers.
Note that both numbers have same base ” a “.
The multiplication of both numbers is given as;
\mathtt{a^{b} \ \times \ a^{c} \ =\ a^{b\ +\ c}}
Note that we have simply added the powers to get the final solution.
Let us look at some examples for further understanding.
Example 01
Multiply \mathtt{5^{3} \ \times \ 5^{4}}
Solution
Note that both the numbers have same base “5”.
So, to multiply the numbers, simply add the exponents.
The multiplication is given as;
\mathtt{\Longrightarrow \ 5^{3} \ \times \ 5^{4} \ }\\\ \\ \mathtt{\Longrightarrow \ 5^{3\ +\ 4}}\\\ \\ \mathtt{\Longrightarrow \ 5^{7}}
On simplification, we get \mathtt{5^{7}}
If you want actual value, then multiply 5 by itself seven times.
\mathtt{\Longrightarrow \ 5^{7}}\\\ \\ \mathtt{\Longrightarrow \ 5\times 5\times 5\times 5\times 5\times 5\times 5}\\\ \\ \mathtt{\Longrightarrow \ 78125}
Hence, 78125 is the final solution.
Example 02
Multiply \mathtt{11^{2} \ \times \ 11^{3} \times 11\ }
Solution
Note that all the given numbers have same base 11.
So, to do the multiplication, simply add the exponents.
\mathtt{\Longrightarrow \ 11^{2} \ \times \ 11^{3} \times 11\ }\\\ \\ \mathtt{\Longrightarrow \ 11^{2\ +\ 3\ +\ 1}}\\\ \\ \mathtt{\Longrightarrow \ 11^{6}}
Hence, \mathtt{11^{6}} is the simplified form of given expression.
Note:
This rule only works if the bases are same.
If we have numbers with different base then we cannot simplify the multiplication by adding the exponents.
Quotient of Power rule for exponents
As multiplication and division are opposite to each other, similarly this rule is opposite to the above mentioned product rule.
According to the law, the division of exponents with the same base can be done by subtraction of exponents.
Let \mathtt{a^{b} \ \&\ a^{c}} are the two numbers.
Note that both the numbers have same base “a”.
Then division of two numbers is given as;
\mathtt{\frac{a^{b}}{a^{c}} =a^{b\ -\ c}}
Note that we have simply subtracted the power to get into the solution.
Let us look at some examples for further understanding.
Example 01
Divide \mathtt{\frac{3^{5}}{3^{2}}}
Solution
Note that both the given numbers have same base.
Simply subtract the exponents to simplify the division.
\mathtt{\Longrightarrow \frac{3^{5}}{3^{2}}}\\\ \\ \mathtt{\Longrightarrow \ 3^{5\ -\ 2}}\\\ \\ \mathtt{\Longrightarrow \ 3^{3}}
Hence, the given division is reduced to number \mathtt{3^{3}}
Example 02
Divide the numbers \mathtt{\frac{13^{25}}{13^{16}}} .
Solution
Note that both the given numbers have same base.
To simplify the division, just subtract the exponents.
\mathtt{\Longrightarrow \frac{13^{25}}{13^{16}}}\\\ \\ \mathtt{\Longrightarrow \ 13^{25\ -\ 16}}\\\ \\ \mathtt{\Longrightarrow \ 13^{9}}
Hence, the division is reduced to \mathtt{13^{9}}
Powers of Power rule
When an exponent is raised by another power then simplification is done by multiplying the exponents.
Consider the exponent \mathtt{\left( a^{b}\right)^{c}} .
Here the exponent \mathtt{a^{b}} is raised to another power c.
According to power of power rule, the simplification can be done my multiplying the exponents.
\mathtt{\left( a^{b}\right)^{c} =\ a^{b\ \times \ c}}
Let us see some examples for better understanding.
Example 01
Solve \mathtt{\left( 2^{5}\right)^{7}}
Solution
Here exponent is raised to another power.
To simplify the expression, simply multiply both the exponents.
\mathtt{\Longrightarrow \ \left( 2^{5}\right)^{7}}\\\ \\ \mathtt{\Longrightarrow \ 2^{5\ \times \ 7}}\\\ \\ \mathtt{\Longrightarrow \ 2^{35}}
Hence, the given number is reduced to \mathtt{2^{35}}
Example 02
Solve \mathtt{\left( 3^{2}\right)^{15}}
Solution
Multiplying the exponents we get;
\mathtt{\Longrightarrow \ \left( 3^{2}\right)^{15}}\\\ \\ \mathtt{\Longrightarrow \ 3^{2\ \times \ 15}}\\\ \\ \mathtt{\Longrightarrow \ 3^{30}}
Hence, \mathtt{\ 3^{30}} is the solution.
Power of Product rule of exponent
The power on a given multiplication can be equally distributed among the multiplication numbers.
For example, if (a x b) is a given product raised to the power n then the power can be distributed to both the numbers a & b.
\mathtt{( a\times b)^{n} =\ a^{n} \times \ b^{n}}
This rule also works in opposite direction.
i.e. the multiplication of numbers with same power can be done as follows;
\mathtt{a^{n} \times \ b^{n} =( a\times b)^{n}}
I hope you understand the above rule. Let us see some examples for further clarity.
Example 01
Multiply \mathtt{2^{3} \times \ 4^{3}}
Solution
Both the given numbers have same exponents.
The multiplication can be done as;
\mathtt{\Longrightarrow 2^{3} \times \ 4^{3}}\\\ \\ \mathtt{\Longrightarrow \ ( 2\times 4)^{3}}\\\ \\ \mathtt{\Longrightarrow \ 8^{3} \ }
Hence, \mathtt{\ 8^{3}} is the solution.
Example 02
Multiply \mathtt{11^{4} \times \ 2^{4}}
Solution
Note that both the multiplication numbers have same exponents.
\mathtt{\Longrightarrow 11^{4} \times \ 2^{4}}\\\ \\ \mathtt{\Longrightarrow \ ( 11\times 2)^{4}}\\\ \\ \mathtt{\Longrightarrow \ 22^{4} \ }
Hence, \mathtt{22^{4} \ } is the solution.
Power of Quotient Rule
According to the exponent law, the power on division of numbers can be equally distributed.
For example;
\mathtt{\left(\frac{a}{b}\right)^{n} =\ a^{n} \div \ b^{n}}
The rule also works in opposite direction.
i.e. the division of number with same power can be done as follows;
\mathtt{\ \frac{a^{n}}{b^{n}} =\left(\frac{a}{b}\right)^{n}}
Let us see some examples for further clarity.
Example 01
Divide the numbers \mathtt{\frac{6^{5}}{3^{5}}}
Solution
Note that both the given numbers have same exponents.
So the division can be done as;
\mathtt{\Longrightarrow \left(\frac{6}{3}\right)^{5}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{\cancel{6} \ \mathbf{2}}{\cancel{3}}\right)^{5}}\\\ \\ \mathtt{\Longrightarrow \ 2^{5}}
Hence, \mathtt{2^{5}} is the solution.
Example 02
Solve \mathtt{\frac{11^{3}}{66^{3}}}
Solution
Note that both the given numbers have same exponents.
The division can be done as;
\mathtt{\Longrightarrow \ \frac{11^{3}}{66^{3}}}\\\ \\ \mathtt{\Longrightarrow \left(\frac{11}{66}\right)^{3}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{\cancel{11} \ }{\cancel{66} \ \mathbf{6}}\right)^{3}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{1}{6}\right)^{3}}
Hence, \mathtt{\left(\frac{1}{6}\right)^{3}} is the solution.
Zero power rule of exponent
According to the exponent law, any number with power 0 will result in number 1.
Some examples of zero power rule are;
\mathtt{a^{0} =1}\\\ \\ \mathtt{5^{0} =1}\\\ \\ \mathtt{\left(\frac{2}{3}\right)^{0} =1}\\\ \\ \mathtt{-17^{0} =1}
Negative exponent rule
Any number with negative exponent result in its reciprocal.
Given below are some examples;
\mathtt{a^{-3} =\frac{1}{a^{3}}}\\\ \\ \mathtt{6^{-4} =\frac{1}{6^{4}}}\\\ \\ \mathtt{\frac{1}{2^{-7}} =\ 2^{7}}
Summary of Rule of Exponents
| Rule | Example |
| Product of Power | \mathtt{a^{m} \times a^{n} =a^{m+n}} |
| Quotient of Power | \mathtt{a^{m} \div a^{n} =\ a^{m-n}} |
| Powers of Power | \mathtt{\left( a^{m}\right)^{n} =a^{m\times n}} |
| Powers of product | \mathtt{( a.b)^{m} =a^{m} \times b^{m}} |
| Power of quotient | \mathtt{\left(\frac{a}{b}\right)^{m} =\ a^{m} \div \ b^{m}} |
| Zero Power rule | \mathtt{a^{0} =1} |
| Negative Power rule | \mathtt{a^{-4} =\frac{1}{a^{4}}} |