Question

select the expression that is equivalent to \\(\sqrt{36x^9y^{12}}\\) given \\(x > 0\\) and \\(y > 0\\).

answer
\\(\circ\\) \\(18x^7y^{10}\\) \\(\circ\\) \\(6x^7y^{10}\\)
\\(\circ\\) \\(18x^{\frac{9}{2}}y^6\\) \\(\circ\\) \\(6x^{\frac{9}{2}}y^6\\)

Explanation:

Step1: Simplify the square root of the constant term

The square root of 36 is 6, so we have $\sqrt{36} = 6$.

Step2: Simplify the square root of the $x$ term

Using the property $\sqrt{a^m} = a^{\frac{m}{2}}$, for $x^9$ we get $x^{\frac{9}{2}}$.

Step3: Simplify the square root of the $y$ term

For $y^{12}$, using the same property, we get $y^{\frac{12}{2}} = y^6$.

Step4: Combine the simplified terms

Putting it all together, $\sqrt{36x^9y^{12}} = 6x^{\frac{9}{2}}y^6$.

Answer:

$6x^{\frac{9}{2}}y^6$ (corresponding to the option: $6x^{\frac{9}{2}}y^6$)