Simplify. $\newline$ Multiply and remove all perfect squares from inside the square roots. Assume $b$ is positive. $\newline$ $\sqrt{24b^{3}}\cdot\sqrt{40b^{2}}\cdot\sqrt{b^{2}}=$

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Algebra 1

Simplify radical expressions with variables

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Q. Simplify.

\newline

Multiply and remove all perfect squares from inside the square roots. Assume

b

is positive.

\newline

\sqrt{24b^{3}}\cdot\sqrt{40b^{2}}\cdot\sqrt{b^{2}}=

Prime Factorization of Square Roots: First, let's find the prime factorization of the numbers inside the square roots and express the variables with exponents that show the perfect squares. $\newline$ $\sqrt{24b^{3}} = \sqrt{2 \cdot 2 \cdot 2 \cdot 3 \cdot b \cdot b \cdot b}$ $\newline$ $\sqrt{40b^{2}} = \sqrt{2 \cdot 2 \cdot 2 \cdot 5 \cdot b \cdot b}$ $\newline$ $\sqrt{b^{2}} = \sqrt{b \cdot b}$
Combining Square Roots: Now, let's combine all the square roots into one square root since the product of square roots is the square root of the product of the numbers. $\newline$ $\sqrt{24b^{3}} \times \sqrt{40b^{2}} \times \sqrt{b^{2}} = \sqrt{24b^{3} \times 40b^{2} \times b^{2}}$
Multiplying Inside the Square Root: Next, multiply the numbers and the variables inside the square root. $\newline$ $\sqrt{24b^{3} \cdot 40b^{2} \cdot b^{2}} = \sqrt{(2 \cdot 2 \cdot 2 \cdot 3) \cdot (2 \cdot 2 \cdot 2 \cdot 5) \cdot (b \cdot b \cdot b) \cdot (b \cdot b) \cdot (b \cdot b)}$ $\newline$ $= \sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 5 \cdot b \cdot b \cdot b \cdot b \cdot b \cdot b \cdot b}$ $\newline$ $= \sqrt{2^{7} \cdot 3 \cdot 5 \cdot b^{7}}$
Removing Perfect Squares: Now, we can remove the perfect squares from inside the square root. $2^6$ is a perfect square, and $b^6$ is a perfect square. $\newline$ $\sqrt{2^7 \cdot 3 \cdot 5 \cdot b^7} = \sqrt{2^6 \cdot 2 \cdot 3 \cdot 5 \cdot b^6 \cdot b}$ $\newline$ $= \sqrt{2^6} \cdot \sqrt{2} \cdot \sqrt{3} \cdot \sqrt{5} \cdot \sqrt{b^6} \cdot \sqrt{b}$ $\newline$ $= 2^3 \cdot \sqrt{2} \cdot \sqrt{3} \cdot \sqrt{5} \cdot b^3 \cdot \sqrt{b}$ $\newline$ $= 8 \cdot \sqrt{2 \cdot 3 \cdot 5 \cdot b}$ $\newline$ $= 8 \cdot \sqrt{30b}$