Question

1. simplify the expression below.

\\(\sqrt3{-15v^5} \cdot \sqrt3{9v^3}\\)

a. \\(-3v^3 \sqrt3{5v}\\)
b. \\(-3v^2 \sqrt3{5v^2}\\)
c. \\(-3v^5 \sqrt3{5}\\)
d. \\(-5v^2 \sqrt3{3v^2}\\)

1. simplify the expression below.

\\(\frac{\sqrt4{240a^{22}}}{\sqrt4{3a^7}}\\)

a. \\(2\sqrt4{5a^3}\\)
b. \\(4\sqrt4{5a^3}\\)
c. \\(2a^3 \sqrt4{5a^3}\\)
d. \\(4a^3 \sqrt4{5a^3}\\)

1. simplify the expression below.

\\(\frac{4\sqrt{5}}{\sqrt{10} - \sqrt{12}}\\)

a. \\(-10\sqrt{2} - 4\sqrt{15}\\)
b. \\(10\sqrt{2} - 4\sqrt{15}\\)
c. \\(-8\sqrt{5} - 2\sqrt{15}\\)
d. \\(-14\sqrt{17}\\)

1. simplify the expression below. write your answer in simplest radical form.

\\(\frac{w^{-\frac{1}{6}} \cdot w^{\frac{8}{3}}}{w^{-1}}\\)

a. \\(w\sqrt{w}\\)
b. \\(\sqrt3{w^2}\\)
c. \\(w^6 \sqrt{w}\\)
d. \\(w^3 \sqrt{w}\\)

Explanation:

Problem 14

Step1: Combine cube roots

$\sqrt[3]{-15v^5 \cdot 9v^3}$

Step2: Multiply inside the root

$\sqrt[3]{-135v^{5+3}} = \sqrt[3]{-135v^8}$

Step3: Factor perfect cubes

$\sqrt[3]{-27 \cdot 5 \cdot v^6 \cdot v^2} = \sqrt[3]{(-3)^3 \cdot 5 \cdot (v^2)^3 \cdot v^2}$

Step4: Extract perfect cubes

$-3v^2\sqrt[3]{5v^2}$

Step1: Combine 4th roots

$\sqrt[4]{\frac{240a^{22}}{3a^7}}$

Step2: Simplify fraction inside root

$\sqrt[4]{80a^{22-7}} = \sqrt[4]{80a^{15}}$

Step3: Factor perfect 4th powers

$\sqrt[4]{16 \cdot 5 \cdot a^{12} \cdot a^3} = \sqrt[4]{2^4 \cdot 5 \cdot (a^3)^4 \cdot a^3}$

Step4: Extract perfect 4th powers

$2a^3\sqrt[4]{5a^3}$

Step1: Rationalize the denominator

$\frac{4\sqrt{5}(\sqrt{10}+\sqrt{12})}{(\sqrt{10}-\sqrt{12})(\sqrt{10}+\sqrt{12})}$

Step2: Expand denominator (difference of squares)

$\frac{4\sqrt{5}(\sqrt{10}+\sqrt{12})}{10-12} = \frac{4\sqrt{5}(\sqrt{10}+\sqrt{12})}{-2}$

Step3: Simplify coefficient

$-2\sqrt{5}(\sqrt{10}+\sqrt{12})$

Step4: Distribute and simplify radicals

$-2(\sqrt{50}+\sqrt{60}) = -2(5\sqrt{2}+2\sqrt{15}) = -10\sqrt{2}-4\sqrt{15}$

Answer:

B. $-3v^2 \sqrt[3]{5v^2}$

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Problem 16