Question

Find the value of m for which \[9^{m} \div 3^{- 2} = 9^{4}\] .

Answer 1

Hint: In this question, we need to find the value of m in the given expression \[9^{m} \div 3^{- 2} = 9^{4}\]. First we can rewrite \[9\] in the form of power, that is , we can rewrite \[9\] as \[3^{2}\] since it is a perfect square number. Then we can use the rules of exponent to make our expression simpler and easier. Important fact is that when the bases are the same , then we can equate the powers .Then on simplifying, we can easily find the value of \[m\].

Complete answer:
Given, \[9^{m} \div 3^{- 2} = 9^{4}\]
First let us rewrite the given expression as
\[\Rightarrow \dfrac{9^{m}}{3^{- 2}} = 9^{4}\]
Now we can rewrite \[9\] as \[3^{2}\] since \[9 = 3 \times 3\] .
On rewriting,
We get,
\[\dfrac{\left( 3^{2} \right)^{m}}{3^{- 2}} = \left( 3^{2} \right)^{4}\]
Now on using the property that \[\dfrac{1}{a^{- n}} = a^{n}\] ,
We get,
\[\left( 3^{2} \right)^{m} \times 3^{2} = \left( 3^{2} \right)^{4}\]
Again on using the property \[\left( a^{m} \right)^{n} = a^{\text{mn}}\],
We get,
\[3^{2m} \times 3^{- 2} = \left( 3^{8} \right)\]
Then on using the property that \[a^{m} + a^{n} = a^{m + n}\],
We get,
\[3^{2m + 2} = 3^{8}\]
Since the bases are equal and same, we can compare the powers,
Now on comparing,
We get,
\[2m + 2 = 8\]
On subtracting both sides by \[2\] ,
We get,
\[\Rightarrow \ 2m = 8 – 2\]
On simplifying,
We get,
\[2m = 6\]
Now on dividing both sides by \[2\] ,
We get,
\[\Rightarrow \ m = \dfrac{6}{2}\]
On simplifying,
We get,
\[m = 3\]
Thus we get the value of \[m\] as \[3\] .
Final answer :
The value of \[m\] is \[3\] .

Note: In order to solve these types of questions, we should have a strong grip over the rule of exponent. Exponent is nothing but a number that is represented in the form \[a^{n}\] . Mathematically there are seven rules of exponent namely Product of powers rule, quotient of powers rule, power of a power rule , power of a product rule ,power of a quotient rule and zero power rule. Also, we should be careful while applying the exponent rules and also while solving it.