Simplify. Assume k is greater than or equal to zero. sqrt 45k^5 ______

Prime Factorization and Expression: sqrt(45k^5) Let's do the prime factors of the radicand for the number 45 and express k^5 in terms of k^2. Prime factorization of a number is the product of its primes. sqrt(45k^5) = sqrt(3 * 3 * 5 * k^4 * k)Group Identical Factors: sqrt(3 * 3 * 5 * k^4 * k) Now, group the identical factors and express k^5 as k^4 times k. Combine the identical factors by using the exponents. sqrt(3 * 3 * 5 * k^4 * k) = sqrt(3^2 * 5 * k^4 * k)Product Property of Radicals: sqrt(3^2 * 5 * k^4 * k) Product property of radicals: sqrt(a * b) = sqrt(a) * sqrt(b) sqrt(3^2 * 5 * k^4 * k) = sqrt(3^2) * sqrt(5) * sqrt(k^4) * sqrt(k)Square and Square Root Cancelation: sqrt(45k^5) = sqrt(3^2) * sqrt(5) * sqrt(k^4) * sqrt(k) Square and square root cancel each other out for 3^2 and k^4. sqrt(3^2) * sqrt(5) * sqrt(k^4) * sqrt(k) = 3 * sqrt(5) * k^2 * sqrt(k)Final Simplification: Final Simplification Since k is greater than or equal to zero, we can leave k^2 and sqrt(k) as they are. The simplified form is 3k^2 * sqrt(5k)

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Simplify. Assume $k$ is greater than or equal to zero. $\newline$ $\sqrt{45k^5}$

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Algebra 1

Simplify radical expressions: mixed review

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Q. Simplify. Assume

k

is greater than or equal to zero.

\newline

\sqrt{45k^5}

Prime Factorization and Expression: $\sqrt{45k^5}$ $\newline$ Let's do the prime factors of the radicand for the number $45$ and express $k^5$ in terms of $k^2$ . $\newline$ Prime factorization of a number is the product of its primes. $\newline$ $\sqrt{45k^5} = \sqrt{3 \times 3 \times 5 \times k^4 \times k}$
Group Identical Factors: $\sqrt{3 \times 3 \times 5 \times k^4 \times k}$ Now, group the identical factors and express $k^5$ as $k^4$ times $k$ . Combine the identical factors by using the exponents. $\sqrt{3 \times 3 \times 5 \times k^4 \times k} = \sqrt{3^2 \times 5 \times k^4 \times k}$
Product Property of Radicals: $\sqrt{3^2 \cdot 5 \cdot k^4 \cdot k}$ $\newline$ Product property of radicals: $\newline$ $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$ $\newline$ $\sqrt{3^2 \cdot 5 \cdot k^4 \cdot k} = \sqrt{3^2} \cdot \sqrt{5} \cdot \sqrt{k^4} \cdot \sqrt{k}$
Square and Square Root Cancelation: $\sqrt{45k^5} = \sqrt{3^2} \cdot \sqrt{5} \cdot \sqrt{k^4} \cdot \sqrt{k}$ Square and square root cancel each other out for $3^2$ and $k^4$ . $\sqrt{3^2} \cdot \sqrt{5} \cdot \sqrt{k^4} \cdot \sqrt{k} = 3 \cdot \sqrt{5} \cdot k^2 \cdot \sqrt{k}$
Final Simplification: Final Simplification $\newline$ Since $k$ is greater than or equal to zero, we can leave $k^2$ and $\sqrt{k}$ as they are. $\newline$ The simplified form is $3k^2 \cdot \sqrt{5k}$