Identify Components: Identify the components of the expression. We have the square root of 75x^(8), which can be written as (75x^(8))^(1/2).Break Down Factors: Break down the expression into prime factors and perfect squares. 75 can be factored into 3 * 5 * 5, and x^(8) is already a perfect square since 8 is an even number. So, (75x^(8))^(1/2) = (3 * 5^2 * x^(8))^(1/2).Apply Square Root: Apply the square root to the factors. The square root of 5^2 is 5, and the square root of x^(8) is x^(8/2) = x^(4). So, (3 * 5^2 * x^(8))^(1/2) = 5x^(4) * (3)^(1/2).Simplify Remaining Root: Simplify the remaining square root. The square root of 3 cannot be simplified further, so we leave it as is. Therefore, the final simplified form is 5x^(4) * sqrt(3).

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$\sqrt{75 x^{8}}$

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Algebra 1

Evaluate integers raised to positive rational exponents

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\sqrt{75 x^{8}}

Identify Components: Identify the components of the expression. $\newline$ We have the square root of $75x^{8}$ , which can be written as $(75x^{8})^{1/2}$ .
Break Down Factors: Break down the expression into prime factors and perfect squares. $\newline$ $75$ can be factored into $3 \times 5 \times 5$ , and $x^{8}$ is already a perfect square since $8$ is an even number. $\newline$ So, $(75x^{8})^{1/2} = (3 \times 5^2 \times x^{8})^{1/2}$ .
Apply Square Root: Apply the square root to the factors. $\newline$ The square root of $5^2$ is $5$ , and the square root of $x^{8}$ is $x^{8/2} = x^{4}$ . $\newline$ So, $(3 \times 5^2 \times x^{8})^{1/2} = 5x^{4} \times (3)^{1/2}$ .
Simplify Remaining Root: Simplify the remaining square root. The square root of $3$ cannot be simplified further, so we leave it as is. Therefore, the final simplified form is $5x^{4} \sqrt{3}$ .