Quotient rule for exponents

In this chapter we will learn about Quotient of power rule of exponents in detail with solved examples.

These rules are important as they are used to solve complex algebra problems in fast and easy way.

Before moving on with this chapter, i would strongly suggest to get your basics clear about the concept of exponents. Click the red link to learn about the topic.

Quotient rule for exponents

According to the rule, the division of exponent with the same base can be done by subtracting the exponents while keeping the original base.

Consider the division of numbers \mathtt{\frac{a^{m}}{a^{n}}}

Note that both the numbers have same base “a”.

The division of exponent rule is expressed as;

\mathtt{\frac{a^{m}}{a^{n}} =a^{m\ -\ n}}

Note that we have simply subtracted the exponents to get the solution.

Can we apply quotient of power rule on numbers with different base ?

NO !!!

The rule is only applicable with number with same base.

Consider the division; \mathtt{\frac{a^{m}}{b^{n}}}

Note that both the numbers have different base. Hence, the quotient rule is not applicable here.

Verification of quotient law of exponents

Consider the division of numbers; \mathtt{\frac{6^{3}}{6^{2}}}

Let us first divide the method using conventional method and find the solution.

\mathtt{\Longrightarrow \ \frac{6^{3}}{6^{2}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{6\times 6\times 6}{6\times 6}}\\\ \\ \mathtt{\Longrightarrow \ \frac{\cancel{6} \times \cancel{6} \times 6}{\cancel{6} \times \cancel{6}}}\\\ \\ \mathtt{\Longrightarrow \ 6}\

Here we got 6 as the solution.

Now solve the expression using quotient rule of exponents.

Note that both the numerator and denominator contain same base, hence the rule applies.

\mathtt{\Longrightarrow \ \frac{6^{3}}{6^{2}}}\\\ \\ \mathtt{\Longrightarrow \ 6^{3-2}}\\\ \\ \mathtt{\Longrightarrow \ 6}

Here we get the same solution as the conventional method.

Hence, the quotient rule is verified.

Different cases of quotient of power rules

Given below are different cases which will encounter while solving division rules of exponents.

(a) Base is fraction

The division rule will still hold even if the given numbers have bases in the form of fraction.

Important is that the number involved have the same base.

\mathtt{\left(\frac{a}{b}\right)^{m} \div \left(\frac{a}{b}\right)^{n} =\left(\frac{a}{b}\right)^{m\ -\ n}}

(b) Base is a negative number

\mathtt{( -a)^{m} \div ( -a)^{n} =( -a)^{m\ -\ n}}

(c) Exponents are negative

The division rule of power will apply to numbers with negative exponents.

The only condition is that the numbers involved have the same base.

\mathtt{\Longrightarrow \ ( a)^{m} \div \ ( a)^{-n}}\\\ \\ \mathtt{\Longrightarrow ( a)^{m\ -( -\ n)}}\\\ \\ \mathtt{\Longrightarrow \ ( a)^{m\ +\ n}}

I hope you understood the above case.

If I want to conclude this division law of exponents in single line, i would say ” Division of number with same base is simplified by subtracting the powers “.

Let us see some examples for further clarity.

Quotient rule for exponents – Solved examples

Example 01
Divide \mathtt{\ \frac{8^{5}}{8^{2}}}

Note that both the numbers have same base.

\mathtt{\Longrightarrow \ \frac{8^{5}}{8^{2}}}\\\ \\ \mathtt{\Longrightarrow \ 8^{5-2}}\\\ \\ \mathtt{\Longrightarrow \ 8^{3}}

Example 02
Simplify the division \mathtt{\frac{11^{7}}{11^{13}}}

Note that both the given number in division have same base. So here we can apply the division rule of exponents.

\mathtt{\Longrightarrow \ \frac{11^{7}}{11^{13}}}\\\ \\ \mathtt{\Longrightarrow \ 11^{7-13}}\\\ \\ \mathtt{\Longrightarrow \ 11^{-6}}

Example 03
Divide \mathtt{\ \left(\frac{2}{3}\right)^{8} \div \ \left(\frac{2}{3}\right)^{2}}

Here both the numbers have same base 2/3.

So, the quotient rule is applicable in this case.

\mathtt{\Longrightarrow \ \left(\frac{2}{3}\right)^{8} \div \ \left(\frac{2}{3}\right)^{2}}\\\ \\ \mathtt{\Longrightarrow \left(\frac{2}{3}\right)^{8\ -\ 2}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{2}{3}\right)^{6}}

Example 04
Simplify the expression \mathtt{\frac{25^{-43}}{25^{-13}}}


\mathtt{\Longrightarrow \ \frac{25^{-43}}{25^{-13}}}\\\ \\ \mathtt{\Longrightarrow \ 25^{-43\ -\ ( -13)}}\\\ \\ \mathtt{\Longrightarrow \ 25^{-43\ +\ 13}}\\\ \\ \mathtt{\Longrightarrow \ 25^{-30}}

Example 05
Simplify \mathtt{\left(\frac{15}{7}\right)^{11} \div \ \left(\frac{15}{7}\right)^{-6}}

\mathtt{\Longrightarrow \ \left(\frac{15}{7}\right)^{11} \div \ \left(\frac{15}{7}\right)^{-6}}\\\ \\ \mathtt{\Longrightarrow \left(\frac{15}{7}\right)^{11\ -\ ( -6)}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{15}{7}\right)^{11\ +\ 6}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{15}{7}\right)^{17}}

Example 06
Simplify \mathtt{\left( xyz^{2}\right)^{-23} \div \ \left( xyz^{2}\right)^{15}}

\mathtt{\Longrightarrow \ \left( xyz^{2}\right)^{-23} \div \ \left( xyz^{2}\right)^{15}}\\\ \\ \mathtt{\Longrightarrow \left( xyz^{2}\right)^{-23\ -\ 15}}\\\ \\ \mathtt{\Longrightarrow \ \left( xyz^{2}\right)^{-38}}

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