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Simplify the radical expression. $\newline$ $\sqrt{12x^{12}}$ $\newline$ Write your answer in the form $A$ , $\sqrt{B}$ , or $A\sqrt{B}$ , where $A$ and $B$ are constants or expressions in $x$ . Use at most one radical in your answer, and at most one absolute value symbol in your expression for $A$ . $\newline$ $\underline{\hspace{3cm}}$

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Simplify radical expressions with variables II

Full solution

Q. Simplify the radical expression.

\newline

\sqrt{12x^{12}}

\newline

Write your answer in the form

A

,

\sqrt{B}

, or

A\sqrt{B}

, where

A

and

B

are constants or expressions in

x

. Use at most one radical in your answer, and at most one absolute value symbol in your expression for

A

.

\newline

\underline{\hspace{3cm}}

Factorizing $12$ : First, we need to factor the number $12$ into its prime factors to see if any of them can be taken out of the square root. $\newline$ $12$ can be factored into $2 \times 2 \times 3$ , which is $2^2 \times 3$ .
Rewriting $x^{12}$ : Next, we look at the variable part, $x^{12}$ . Since the exponent is even, we can rewrite $x^{12}$ as $(x^{6})^{2}$ , which is a perfect square.
Rewriting the expression: Now we can rewrite the original expression using these factors: $\newline$ $\sqrt{12x^{12}} = \sqrt{2^2 \cdot 3 \cdot (x^6)^2}$ .
Taking the square root: We can take the square root of the perfect squares, which are $2^2$ and $(x^6)^2$ , and move them outside the square root. The square root of $2^2$ is $2$ , and the square root of $(x^6)^2$ is $x^6$ . $\newline$ So, we have $2x^6 \cdot \sqrt{3}$ .
Final simplified expression: The expression $2x^6 \sqrt{3}$ is already simplified and in the form $A\sqrt{B}$ , where $A$ is $2x^6$ and $B$ is $3$ . There is no need for an absolute value symbol because $x^6$ is always non-negative.

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