Question

Simplify the given fraction: $\dfrac{{{3^7}}}{{{3^2}}}$

Answer 1

We can use the properties of indices to solve this expression in a simpler manner. The property dealing with fractional indices which is used here is following $\dfrac{{{p^q}}}{{{p^r}}} = {p^{q - r}}$. We use this property to obtain our result.

Complete solution step by step:
Firstly we write down the expression
$\dfrac{{{3^7}}}{{{3^2}}}$
This expression is in the fractional form and we know that the fraction consists of two parts i.e. numerator and denominator. It is expressed as – numerator upon denominator. And in our question the expression also has indices (powers) on both parts of it and also-
It is written in the standard form of index property which is given by this formula -
$\dfrac{{{p^q}}}{{{p^r}}} = {p^{q - r}}$
Here the powers of numerator and denominator are subtracted respectively to get to the result i.e.
$\dfrac{{{3^7}}}{{{3^2}}} = {3^{7 - 2}} = {3^5}$
Now we have obtained a value which has an index of ‘5’ placed over it so we have to solve it by multiplying the number ‘3’ by itself five times to get to our final result
$\Rightarrow {3^5} = 3 \times 3 \times 3 \times 3 \times 3 \\\ = 9 \times 9 \times 3 \\\ = 243 \\\$
So we have simplified the value of the given expression that is ‘243’.
Additional information: We can solve this question without using the index property and use here simple multiplication division method like this
$\dfrac{{{3^7}}}{{{3^2}}} = \dfrac{{3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3}}{{3 \times 3}} \\\ = 3 \times 3 \times 3 \times 3 \times 3 \\\ = 243 \\\$
We have ‘243’ as our answer by solving this way.

Note: The method used in the first part of the solution is more appropriate to solve the question because the second part is time taking and long too. Using the property in the first part we can reduce the calculation and make it easier.