Identify Expression: Identify the expression to be simplified. We have sqrt(12v^(6)).Break Down Square Root: Break down the square root of the product into the product of square roots. sqrt(12v^(6)) can be written as sqrt(12) * sqrt(v^(6)).Simplify Square Root of 12: Simplify sqrt(12) by finding the prime factors of 12. The prime factorization of 12 is 2 * 2 * 3, which can be written as 2^2 * 3.Extract Perfect Square Root: Extract the square root of the perfect square from sqrt(12). Since 2^2 is a perfect square, sqrt(2^2 * 3) equals 2 * sqrt(3).Simplify Square Root of v^6: Simplify sqrt(v^(6)) by using the property of square roots that sqrt(a^2) = a. Since v^(6) is a perfect square ((v^3)^2), sqrt(v^(6)) equals v^3.Combine Simplified Parts: Combine the simplified parts to get the final answer. Multiplying 2 * sqrt(3) by v^3 gives 2v^3 * sqrt(3).

Precalculus

Evaluate rational exponents

Full solution

\sqrt{12 v^{6}}

Identify Expression: Identify the expression to be simplified. $\newline$ We have $\sqrt{12v^{6}}$ .
Break Down Square Root: Break down the square root of the product into the product of square roots. $\sqrt{12v^{6}}$ can be written as $\sqrt{12} \times \sqrt{v^{6}}$ .
Simplify Square Root of $12$ : Simplify $\sqrt{12}$ by finding the prime factors of $12$ . $\newline$ The prime factorization of $12$ is $2 \times 2 \times 3$ , which can be written as $2^2 \times 3$ .
Extract Perfect Square Root: Extract the square root of the perfect square from $\sqrt{12}$ . $\newline$ Since $2^2$ is a perfect square, $\sqrt{2^2 \times 3}$ equals $2 \times \sqrt{3}$ .
Simplify Square Root of $v^6$ : Simplify $\sqrt{v^{6}}$ by using the property of square roots that $\sqrt{a^2} = a$ . Since $v^{6}$ is a perfect square ( $(v^3)^2$ ), $\sqrt{v^{6}}$ equals $v^3$ .
Combine Simplified Parts: Combine the simplified parts to get the final answer. $\newline$ Multiplying $2 \times \sqrt{3}$ by $v^3$ gives $2v^3 \times \sqrt{3}$ .