Power of quotient rule

In this chapter, we will discuss power of quotient rule with solved examples.

we have already discussed multiplication rule of exponents in different chapter. Click the red link to read about the same.

Power of quotient rule

According to the property, the common exponent on the division of numbers can be equally distributed among the numerator and denominator.

The rule can be expressed as;

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

In the above expression, you can see that in fraction (a/b), the power “m” is individually distributed among numerator and denominator.

The opposite of above expression is also true. If the numerator and denominator has same power then we can combine both the powers into one.

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

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

Power of quotient rule for exponents – Solved examples

Example 01
Simplify \mathtt{6^{3} \div 2^{3}}

Note that both the numbers have same power.

Applying Power of quotient rule for exponents.

\mathtt{\Longrightarrow \ 6^{3} \div 2^{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{6^{3}}{2^{3}}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{6}{2}\right)^{3}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{\cancel{6} \ \mathbf{3}}{\cancel{2}}\right)^{3}}\\\ \\ \mathtt{\Longrightarrow \ 3^{3}}\\\ \\ \mathtt{\Longrightarrow \ 27}

Example 02
Simplify \mathtt{35^{10} \div 7^{10}}

Note that both the dividing numbers have same exponents.

Applying power of quotient rule, we get;

\mathtt{\Longrightarrow \ 35^{10} \div 7^{10}}\\\ \\ \mathtt{\Longrightarrow \ \frac{35^{10}}{7^{10}}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{35}{7}\right)^{10}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{\cancel{35} \ 5}{\cancel{7}}\right)^{10}}\\\ \\ \mathtt{\Longrightarrow \ 5^{10}}

Example 03
Find the value of \mathtt{\left(\frac{3}{2}\right)^{4}}

Using power of quotient rule, we can individually distribute power on numerator and denominator.

\mathtt{\Longrightarrow \left(\frac{3}{2}\right)^{4}}\\\ \\ \mathtt{\Longrightarrow \ \frac{3^{4}}{2^{4}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{81}{16}}

Example 04
Find the value of division \mathtt{\left(\frac{-5}{8}\right)^{3}}


\mathtt{\Longrightarrow \left(\frac{-5}{8}\right)^{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-5^{3}}{8^{3}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-5\times -5\times -5}{8\times 8\times 8}}\\\ \\ \mathtt{\Longrightarrow \frac{-125}{512}}

Example 05
Find the value of exponent \mathtt{\left(\frac{3}{7}\right)^{-2}}

Here the number contains negative exponent.

Take reciprocal of number to convert the exponent from negative to positive and then apply power of quotient rule.

\mathtt{\Longrightarrow \left(\frac{3}{7}\right)^{-2}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{7}{3}\right)^{2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{7^{2}}{3^{2}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{7\times 7}{3\times 3}}\\\ \\ \mathtt{\Longrightarrow \frac{49}{9}}

Example 06
Simplify \mathtt{-9^{23} \div 13^{23}}

\mathtt{\Longrightarrow \ -9^{23} \div 13^{23}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-9^{23}}{13^{23}}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{-9}{13}\right)^{23}}

Example 07
Simplify \mathtt{16^{-120} \div 27^{-120}}

Simplifying the expression.

\mathtt{\Longrightarrow \ 16^{-120} \div 27^{-120}}\\\ \\ \mathtt{\Longrightarrow \ \frac{16^{-120}}{27^{-120}}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{16}{27}\right)^{-120}}

Take the reciprocal to convert the negative power into positive.

\mathtt{\Longrightarrow \ \left(\frac{16}{27}\right)^{-120}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{27}{16}\right)^{120}}

Example 08
Find the value of \mathtt{\left(\frac{-11}{5}\right)^{3}}

\mathtt{\Longrightarrow \ \left(\frac{-11}{5}\right)^{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-11^{3}}{5^{3}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-11\times -11\times -11}{5\times 5\times 5}}\\\ \\ \mathtt{\Longrightarrow \frac{-1331}{125}}

Example 09
Simplify \mathtt{\left(\frac{5x}{x+y}\right)^{2}}


\mathtt{\Longrightarrow \ \left(\frac{5x}{x+y}\right)^{2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{( 5x)^{2}}{( x+y)^{2}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{25x^{2}}{x^{2} +y^{2} +2xy}}

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