Identify Components: Identify the components of the expression sqrt(75a^(6)). In sqrt(75a^(6)), we have a square root of a product of a number and a variable raised to a power.Factor Number: Factor the number 75 into prime factors. 75 can be factored into 3 * 5 * 5.Rewrite Expression: Rewrite the expression using the prime factorization of 75. sqrt(75a^(6)) becomes sqrt(3 * 5^2 * a^(6)).Simplify Square Root: Simplify the square root by taking out factors that are perfect squares. Since 5^2 is a perfect square, it can be taken out of the square root as 5. The variable a^(6) is also a perfect square because 6 is an even number, so a^(6) can be taken out of the square root as a^(3). sqrt(3 * 5^2 * a^(6)) becomes 5a^(3) * sqrt(3).Write Final Expression: Write the final simplified expression. The simplified form of sqrt(75a^(6)) is 5a^(3) * sqrt(3).

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$\sqrt{75 a^{6}}$

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Algebra 2

Evaluate rational exponents

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\sqrt{75 a^{6}}

Identify Components: Identify the components of the expression $\sqrt{75a^{6}}$ . In $\sqrt{75a^{6}}$ , we have a square root of a product of a number and a variable raised to a power.
Factor Number: Factor the number $75$ into prime factors. $75$ can be factored into $3 \times 5 \times 5$ .
Rewrite Expression: Rewrite the expression using the prime factorization of $75$ . $\sqrt{75a^{6}}$ becomes $\sqrt{3 \times 5^2 \times a^{6}}$ .
Simplify Square Root: Simplify the square root by taking out factors that are perfect squares. $\newline$ Since $5^2$ is a perfect square, it can be taken out of the square root as $5$ . The variable $a^{6}$ is also a perfect square because $6$ is an even number, so $a^{6}$ can be taken out of the square root as $a^{3}$ . $\newline$ $\sqrt{3 \times 5^2 \times a^{6}}$ becomes $5a^{3} \times \sqrt{3}$ .
Write Final Expression: Write the final simplified expression. $\newline$ The simplified form of $\sqrt{75a^{6}}$ is $5a^{3} \times \sqrt{3}$ .