Question

How do you simplify $\sqrt {{x^6}{y^6}} .$

Answer 1

Hint:For simplify the expressions containing square root we have to remove all perfect squares if any present inside the square root. So here we have to factorize the given expression in terms of perfect squares and then simplify it.

Complete step by step solution:
Given
$\sqrt {{x^6}{y^6}} ......................................\left( i \right)$

Now we have to change the expression${x^6}{y^6}$into a one of perfect squares such that we can take one of the terms outside the square root.

Also we know the exponential property:
${x^m} \times {x^n} = {x^{m + n}}........................\left( {ii} \right)$

So by using property stated in (ii) we have to simplify the expression ${x^6}{y^6}$and thereby$\sqrt
{{x^6}{y^6}} $.

So simplifying${x^6}{y^6}$:

We know that by using (ii) we can write:
\[{x^6} = {x^3} \times {x^3}......................\left( {iii} \right)\]

Similarly we can write:
${y^6} = {y^3} \times {y^3}.................................\left( {iv} \right)$

So we were able to express ${x^6}{y^6}$as products of perfect square, now we have to substitute this back in (i) and simplify further:

\[ \Rightarrow \sqrt {{x^6}{y^6}} = \sqrt {\left( {{x^3} \times {x^3}} \right) \times \left( {{y^3} \times {y^3}} \right)} ................\left( v \right)\]

Now on observing (v) we can say that inside the square root the terms are expressed in the form of perfect squares and therefore we can take one of the terms outside the root.

Such that:
\[ \Rightarrow \sqrt {\left( {{x^3} \times {x^3}} \right) \times \left( {{y^3} \times {y^3}} \right)} =
{x^3} \times {y^3}......\left( {vi} \right)\]

Therefore from (vi) we can write that on simplifying $\sqrt {{x^6}{y^6}} $we get:
$\sqrt {{x^6}{y^6}} = {x^3} \times {y^3} = {x^3}{y^3}$

Note: Whenever questions including exponents are given some of the identities useful are:
$
{x^m} \times {x^n} = {\left( x \right)^{m + n}} \\
\dfrac{{{x^n}}}{{{x^m}}} = {\left( x \right)^{n - m}} \\
{\left( {{x^n}} \right)^m} = {\left( x \right)^{n \times m}} \\
$
So our given expressions should be converted and expressed based on the above standard identities, by which it would be much easier to simplify and solve it. Also radical expressions are algebraic expressions which have or contain radicals, and the best way to solve a square root is to remove all the perfect squares from inside the square root if any exists.