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Simplify. Assume $y$ is greater than or equal to zero. $\newline$ $\sqrt{75y^7}$

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Simplify radical expressions with variables

Full solution

Q. Simplify. Assume

y

is greater than or equal to zero.

\newline

\sqrt{75y^7}

Factorize $75y^7$ : Factor $75y^7$ into its prime factors. $\newline$ The prime factorization of $75$ is $3 \times 5 \times 5$ . Since $y^7$ is already a power of a prime number, we can write $75y^7$ as $3 \times 5^2 \times y^7$ .
Group Factors: Group the factors under the square root into pairs of perfect squares. $\newline$ We can group the factors as follows: $\sqrt{3 \times 5^2 \times y^6 \times y}$ . $\newline$ Here, $5^2$ and $y^6$ are perfect squares.
Simplify Perfect Squares: Simplify the square root of the perfect squares. $\newline$ The square root of a perfect square is the number that was squared. So, $\sqrt{5^2} = 5$ and $\sqrt{y^6} = y^3$ .
Take Out Squares: Take the perfect squares out of the square root. We can now take the square root of the perfect squares out of the radical, which gives us $5y^3 \times \sqrt{3y}$ .
Write Final Expression: Write the final simplified expression. $\newline$ The final simplified expression is $5y^3 \times \sqrt{3y}$ .

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