Question

simplify the radical expressions. (answers should be in simplest radical form.)

1. $6\sqrt{135}$
2. $- 3\sqrt3{112}$
3. $\sqrt{75x^{3}y^{4}}$

$\sqrt3{192x^{7}y^{3}}$

1. $\sqrt5{96x^{6}y^{3}}$
2. $3x\sqrt3{48x^{7}}$

$\frac{7}{3}x^{3}y^{2}\sqrt4{162xy^{5}}$

1. $\frac{1}{2}\sqrt5{160x^{7}}$
2. $ - 2\sqrt3{48}+4\sqrt3{162}$

Explanation:

Problem 7: Simplify $6\sqrt[4]{135}$

Step1: Factor radicand

$135 = 3^3 \times 5$

Step2: Simplify 4th root

$\sqrt[4]{135} = \sqrt[4]{81 \times \frac{135}{81}} = \sqrt[4]{3^4 \times \frac{5}{3}} = 3\sqrt[4]{\frac{5}{3}} = 3\sqrt[4]{\frac{5 \times 3^3}{3^4}} = 3 \times \frac{\sqrt[4]{135}}{3}$ → Correct factor: $135=3^3×5$, no 4th power factors, so $\sqrt[4]{135}$ stays.

Step3: Multiply by coefficient

$6\sqrt[4]{135}$ (no perfect 4th power factors in 135, so this is simplified)

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Problem 8: Simplify $-3\sqrt[4]{112}$

Step1: Factor radicand

$112 = 16 \times 7 = 2^4 \times 7$

Step2: Simplify 4th root

$\sqrt[4]{112} = \sqrt[4]{2^4 \times 7} = 2\sqrt[4]{7}$

Step3: Multiply by coefficient

$-3 \times 2\sqrt[4]{7} = -6\sqrt[4]{7}$

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Problem 9: Simplify $\sqrt{75x^3y^4}$

Step1: Factor radicand

$75x^3y^4 = 25 \times 3 \times x^2 \times x \times (y^2)^2 = 5^2 x^2 (y^2)^2 \times 3x$

Step2: Simplify square root

$\sqrt{5^2 x^2 (y^2)^2 \times 3x} = 5x y^2 \sqrt{3x}$

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Problem 10: Simplify $\sqrt[3]{192x^7y^3}$

Step1: Factor radicand

$192x^7y^3 = 64 \times 3 \times x^6 \times x \times y^3 = 4^3 (x^2)^3 y^3 \times 3x$

Step2: Simplify cube root

$\sqrt[3]{4^3 (x^2)^3 y^3 \times 3x} = 4x^2 y \sqrt[3]{3x}$

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Problem 11: Simplify $\sqrt[5]{96x^6y^3}$

Step1: Factor radicand

$96x^6y^3 = 32 \times 3 \times x^5 \times x \times y^3 = 2^5 x^5 \times 3x y^3$

Step2: Simplify 5th root

$\sqrt[5]{2^5 x^5 \times 3x y^3} = 2x \sqrt[5]{3x y^3}$

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Problem 12: Simplify $3x\sqrt[3]{48x^7}$

Step1: Factor radicand

$48x^7 = 8 \times 6 \times x^6 \times x = 2^3 (x^2)^3 \times 6x$

Step2: Simplify cube root

$\sqrt[3]{2^3 (x^2)^3 \times 6x} = 2x^2 \sqrt[3]{6x}$

Step3: Multiply by coefficient

$3x \times 2x^2 \sqrt[3]{6x} = 6x^3 \sqrt[3]{6x}$

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Problem 13: Simplify $\frac{7}{3}x^3y^2\sqrt[4]{162xy^5}$

Step1: Factor radicand

$162xy^5 = 81 \times 2 \times x \times y^4 \times y = 3^4 y^4 \times 2x y$

Step2: Simplify 4th root

$\sqrt[4]{3^4 y^4 \times 2x y} = 3y \sqrt[4]{2x y}$

Step3: Multiply by coefficient

$\frac{7}{3}x^3y^2 \times 3y \sqrt[4]{2x y} = 7x^3y^3 \sqrt[4]{2xy}$

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Problem 14: Simplify $\frac{1}{2}\sqrt[5]{160x^7}$

Step1: Factor radicand

$160x^7 = 32 \times 5 \times x^5 \times x^2 = 2^5 x^5 \times 5x^2$

Step2: Simplify 5th root

$\sqrt[5]{2^5 x^5 \times 5x^2} = 2x \sqrt[5]{5x^2}$

Step3: Multiply by coefficient

$\frac{1}{2} \times 2x \sqrt[5]{5x^2} = x\sqrt[5]{5x^2}$

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Problem 15: Simplify $-2\sqrt[3]{48} + 4\sqrt[3]{162}$

Step1: Factor first radicand

$48 = 8 \times 6 = 2^3 \times 6$

Step2: Simplify first cube root

$\sqrt[3]{48} = 2\sqrt[3]{6}$

Step3: Factor second radicand

$162 = 27 \times 6 = 3^3 \times 6$

Step4: Simplify second cube root

$\sqrt[3]{162} = 3\sqrt[3]{6}$

Step5: Substitute and combine terms

$-2 \times 2\sqrt[3]{6} + 4 \times 3\sqrt[3]{6} = -4\sqrt[3]{6} + 12\sqrt[3]{6} = 8\sqrt[3]{6}$

Answer:

1. $\boldsymbol{6\sqrt[4]{135}}$
2. $\boldsymbol{-6\sqrt[4]{7}}$
3. $\boldsymbol{5xy^2\sqrt{3x}}$
4. $\boldsymbol{4x^2y\sqrt[3]{3x}}$
5. $\boldsymbol{2x\sqrt[5]{3xy^3}}$
6. $\boldsymbol{6x^3\sqrt[3]{6x}}$
7. $\boldsymbol{7x^3y^3\sqrt[4]{2xy}}$
8. $\boldsymbol{x\sqrt[5]{5x^2}}$
9. $\boldsymbol{8\sqrt[3]{6}}$