Question

How do you solve \[\sqrt {243{x^8}{y^5}} \] ?

Answer 1

Hint: The given question has a polynomial. Now the root sign signifies that we need to get the perfect square terms from the polynomial. So we will write 243 as a product of 81 and 3 since 81 is a perfect square of 9. And in case of the variables we will use the laws of powers.

Complete step by step solution:
Given that \[\sqrt {243{x^8}{y^5}} \]
Now we can write \[243 = 81 \times 3\]
\[{x^8} = {\left( {{x^4}} \right)^2}\] because we know that \[{x^{mn}} = \left( {{{\left( {{x^m}} \right)}^n}} \right)\]
So now the above term become,
$\Rightarrow$\[\sqrt {243{x^8}{y^5}} = \sqrt {81 \times 3{{\left( {{x^4}} \right)}^2}{y^5}} \]
On proceeding further, we can write
$\Rightarrow$\[\sqrt {243{x^8}{y^5}} = 9\sqrt {3{{\left( {{x^4}} \right)}^2}{y^5}} \]
Now for the term of x we can write,
$\Rightarrow$\[\sqrt {243{x^8}{y^5}} = 9{\left( {{{\left( {{x^4}} \right)}^2}} \right)^{\dfrac{1}{2}}}\sqrt {3{y^5}} \]
$\Rightarrow$\[\sqrt {243{x^8}{y^5}} = 9{x^4}\sqrt {3{y^5}} \]
This is the correct answer.

Note: Here note that the simplified answer should be written as your final answer. Here the variable y is in the root because though we used laws of indices, we cannot simplify it further because the power is not a multiple of 2. So keep it as it is. And in case of x we converted it to simplify because the power was a multiple of 2.