Question

How do you simplify $\dfrac{{\dfrac{{10ab}}{{{x^2} - {y^2}}}}}{{\dfrac{{5a(x - y)}}{{3ax(x + y)}}}}$?

Answer 1

Hint: First of all we will convert the given expression in the simple form and then simplest form. Numerator’s denominator goes to the denominator and the denominator’s denominator goes to the numerator.

Complete step by step solution:
Take the given expression: $\dfrac{{\dfrac{{10ab}}{{{x^2} - {y^2}}}}}{{\dfrac{{5a(x - y)}}{{3ax(x + y)}}}}$
Numerator’s denominator goes to the denominator and the denominator’s denominator goes to the numerator.
$ = \dfrac{{10ab}}{{5a(x - y)}} \times \dfrac{{3ax(x + y)}}{{{x^2} - {y^2}}}$
Common factors from the numerator and the denominator cancels each other. Therefore remove “a” from the numerator and the denominator. Also find the factors of the terms such as $10 = 2 \times 5$and using the identity of difference of two squares ${x^2} - {y^2} = (x - y)(x + y)$
$ = \dfrac{{2 \times 5b}}{{5(x - y)}} \times \dfrac{{3ax(x + y)}}{{(x + y)(x - y)}}$
Now, remove the common factors $5,\;{\text{(x + y)}}$from the above expression –
$ = \dfrac{{2b}}{{(x - y)}} \times \dfrac{{3ax}}{{(x - y)}}$
The above expression can be re-written as –
$ = \dfrac{{6abx}}{{{{(x - y)}^2}}}$
This is the required solution.

Note: Be good in finding the factors of the terms. Always remember that the common factors from the numerator and the denominator cancels each other.
Prime factorization is the process of finding which prime numbers can be multiplied together to make the original number, where prime numbers are the numbers greater than $1$ and which are not the product of any two smaller natural numbers. For Example: $2,{\text{ 3, 5, 7,}}......$ $2$ is the prime number as it can have only $1$ factor.