Simplify. sqrt 54 ______

Find Prime Factors: sqrt(54) Let's start by finding the prime factors of the radicand (the number inside the square root). Prime factorization of a number is expressing it as the product of its prime factors. sqrt(54) = sqrt(2 * 3 * 3 * 3)Group and Combine Factors: sqrt(2 * 3 * 3 * 3) Now, group the identical factors. Combine the identical factors by using the exponents. sqrt(2 * 3 * 3 * 3) = sqrt(2 * 3^3)Apply Product Property: sqrt(2 * 3^3) Product property of radicals: sqrt(a * b) = sqrt(a) * sqrt(b) sqrt(2 * 3^3) = sqrt(2) * sqrt(3^3)Separate Perfect Square: sqrt(2) * sqrt(3^3) Since 3^3 is 3 * 3 * 3, we can take out a pair of 3s from under the square root as a single 3. sqrt(2) * sqrt(3^3) = sqrt(2) * sqrt(3^2 * 3)Cancel Square Root: sqrt(2) * sqrt(3^2 * 3) Now, apply the product property of radicals to separate the perfect square 3^2 from 3. sqrt(2) * sqrt(3^2 * 3) = sqrt(2) * sqrt(3^2) * sqrt(3)Multiply Outside Number: sqrt(2) * sqrt(3^2) * sqrt(3) Square and square root cancel each other out for sqrt(3^2). sqrt(2) * sqrt(3^2) * sqrt(3) = sqrt(2) * 3 * sqrt(3)Combine Square Roots: sqrt(2) * 3 * sqrt(3) Since sqrt(2) and sqrt(3) cannot be simplified further, we multiply the outside number 3 by sqrt(2) and sqrt(3) separately. sqrt(2) * 3 * sqrt(3) = 3 * sqrt(2) * sqrt(3)Multiply Numbers Under Square Root: 3 * sqrt(2) * sqrt(3) Combine the square roots under a single radical. 3 * sqrt(2) * sqrt(3) = 3 * sqrt(2 * 3)3 * sqrt(2 * 3) Now, multiply the numbers under the square root. 3 * sqrt(2 * 3) = 3 * sqrt(6)

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Algebra 1

Simplify radical expressions: mixed review

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Q. Simplify.

\newline

\sqrt{54}

Find Prime Factors: $\sqrt{54}$ $\newline$ Let's start by finding the prime factors of the radicand (the number inside the square root). $\newline$ Prime factorization of a number is expressing it as the product of its prime factors. $\newline$ $\sqrt{54} = \sqrt{2 \times 3 \times 3 \times 3}$
Group and Combine Factors: $\sqrt{2 \times 3 \times 3 \times 3}$ $\newline$ Now, group the identical factors. $\newline$ Combine the identical factors by using the exponents. $\newline$ $\sqrt{2 \times 3 \times 3 \times 3} = \sqrt{2 \times 3^3}$
Apply Product Property: $\sqrt{2 \times 3^3}$ $\newline$ Product property of radicals: $\newline$ $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$ $\newline$ $\sqrt{2 \times 3^3} = \sqrt{2} \times \sqrt{3^3}$
Separate Perfect Square: $\sqrt{2} \times \sqrt{3^3}$ $\newline$ Since $3^3$ is $3 \times 3 \times 3$ , we can take out a pair of $3$ s from under the square root as a single $3$ . $\newline$ $\sqrt{2} \times \sqrt{3^3} = \sqrt{2} \times \sqrt{3^2 \times 3}$
Cancel Square Root: $\sqrt{2} \times \sqrt{3^2 \times 3}$ $\newline$ Now, apply the product property of radicals to separate the perfect square $3^2$ from $3$ . $\newline$ $\sqrt{2} \times \sqrt{3^2 \times 3} = \sqrt{2} \times \sqrt{3^2} \times \sqrt{3}$
Multiply Outside Number: $\sqrt{2} \times \sqrt{3^2} \times \sqrt{3}$ $\newline$ Square and square root cancel each other out for $\sqrt{3^2}$ . $\newline$ $\sqrt{2} \times \sqrt{3^2} \times \sqrt{3} = \sqrt{2} \times 3 \times \sqrt{3}$
Combine Square Roots: $\sqrt{2} \times 3 \times \sqrt{3}$ $\newline$ Since $\sqrt{2}$ and $\sqrt{3}$ cannot be simplified further, we multiply the outside number $3$ by $\sqrt{2}$ and $\sqrt{3}$ separately. $\newline$ $\sqrt{2} \times 3 \times \sqrt{3} = 3 \times \sqrt{2} \times \sqrt{3}$
Multiply Numbers Under Square Root: $3 \times \sqrt{2} \times \sqrt{3}$ Combine the square roots under a single radical. $3 \times \sqrt{2} \times \sqrt{3} = 3 \times \sqrt{2 \times 3}$
Multiply Numbers Under Square Root: $3 \times \sqrt{2} \times \sqrt{3}$ $\newline$ Combine the square roots under a single radical. $\newline$ $3 \times \sqrt{2} \times \sqrt{3} = 3 \times \sqrt{2 \times 3}3 \times \sqrt{2 \times 3}$ $\newline$ Now, multiply the numbers under the square root. $\newline$ $3 \times \sqrt{2 \times 3} = 3 \times \sqrt{6}$