Question

Which number is equivalent to $\dfrac{{{3}^{4}}}{{{3}^{2}}}$ ?

Answer 1

Hint: Here we have been given a fraction between two exponents and we have to find its value. Firstly as we can see that both numerator and denominator term have exponent with same base but power of them is different so we will use the formula $\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}$ and simplify our value. Then we will simplify our value further by using exponent definition and get our desired answer.

Complete answer:
We have to evaluate the below value:
$\dfrac{{{3}^{4}}}{{{3}^{2}}}$….$\left( 1 \right)$
Now as we can see that exponent in both numerator and denominator has same base so we can use the below formula:
$\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}$
Comparing the above formula by equation (1) we get,
$a=3,m=4,n=2$
Substituting the above value in the formula we get,
$\Rightarrow \dfrac{{{3}^{4}}}{{{3}^{2}}}={{3}^{4-2}}$
$\Rightarrow \dfrac{{{3}^{4}}}{{{3}^{2}}}={{3}^{2}}$
By exponent definition ${{a}^{n}}=a\times a\times a\times a.....n\,times$ so,
$\Rightarrow \dfrac{{{3}^{4}}}{{{3}^{2}}}=3\times 3$
$\Rightarrow \dfrac{{{3}^{4}}}{{{3}^{2}}}=9$
Hence the number $9$ is equivalent to $\dfrac{{{3}^{4}}}{{{3}^{2}}}$ .

Note:
Exponent and power are used to represent large number in simplified and compact form. If we are multiplying any number $18$ times by itself writing such big value is complicated and chance of error is more so in this case we can use exponent where we can directly write the number to the power $18$ .There are many law of exponent such as division law, multiplication law and negative exponent law which is very important while solving exponent questions. This question can be done in other way also where we can simplify the numerator and denominator by using exponent definition and then we can divide them to get our answer as follows:
$\Rightarrow \dfrac{{{3}^{4}}}{{{3}^{2}}}$
By exponent definition the base number is multiplied by itself the times of power so,
$\Rightarrow \dfrac{3\times 3\times 3\times 3}{3\times 3}$
$\Rightarrow \dfrac{81}{9}$
On simplifying we get,
$\Rightarrow 9$
This is the same as our above answer.