Question

question
express in simplest radical form assuming ( x > 0 ).
( sqrt{36x^{11}} )
answer
( sqrt{36x^{11}} ) ( 6sqrt{x^{11}} )
( 6x^{5}sqrt{x} ) ( x^{5}sqrt{36x} )

Explanation:

Step1: Simplify the square root of 36

The square root of 36 is 6, so we can rewrite $\sqrt{36x^{11}}$ as $\sqrt{36} \cdot \sqrt{x^{11}} = 6\sqrt{x^{11}}$.

Step2: Simplify the square root of $x^{11}$

We can express $x^{11}$ as $x^{10} \cdot x$, since $x^{10} \cdot x = x^{10 + 1} = x^{11}$. Then $\sqrt{x^{11}} = \sqrt{x^{10} \cdot x}$.
Using the property of square roots $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$ (where $a = x^{10}$ and $b = x$), we get $\sqrt{x^{10}} \cdot \sqrt{x}$.
Since $\sqrt{x^{10}} = x^5$ (because $(x^5)^2 = x^{10}$ and $x > 0$), then $\sqrt{x^{10}} \cdot \sqrt{x} = x^5\sqrt{x}$.

Step3: Combine the results

From Step1, we have $6\sqrt{x^{11}}$, and from Step2, we know $\sqrt{x^{11}} = x^5\sqrt{x}$. So substituting back, we get $6 \cdot x^5\sqrt{x} = 6x^5\sqrt{x}$.

Answer:

$6x^5\sqrt{x}$ (corresponding to the option "6x⁵√x")