Question

6.4 classwork/homework: multiplying & dividing nth root radicals

1. $2\sqrt{6xy} \cdot 3x\sqrt{3x^2y}$
2. $\sqrt{\frac{72x^5y^3}{4x^2y}}$
3. $\frac{\sqrt{144x^5}}{3\sqrt{2x}}$
4. $\sqrt4{8xy^3} \cdot \sqrt4{6x^3y}$
5. $9y\sqrt{12x} \cdot x^2\sqrt{10xy^3}$
6. $\sqrt4{\frac{80xy^9}{5xy}}$
7. $\frac{7\sqrt3{320x^{11}y^5}}{2\sqrt3{5x^8y^5}}$
8. $3\sqrt3{3x^4} \cdot \sqrt3{9x^2}$
9. $-3\sqrt3{4x^2} \cdot 5\sqrt3{16x}$
10. $\frac{3\sqrt{192x^6y^4}}{4\sqrt{3x^3y}}$

Explanation:

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Problem 1

Step1: Multiply coefficients & radicands

$2 \cdot 3x \cdot \sqrt{6xy \cdot 3x^2y} = 6x\sqrt{18x^3y^2}$

Step2: Simplify the radical

$\sqrt{18x^3y^2} = \sqrt{9 \cdot 2 \cdot x^2 \cdot x \cdot y^2} = 3xy\sqrt{2x}$

Step3: Multiply remaining terms

$6x \cdot 3xy\sqrt{2x} = 18x^2y\sqrt{2x}$

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Problem 2

Step1: Simplify fraction inside radical

$\sqrt{\frac{72x^5y^3}{4x^2y}} = \sqrt{18x^3y^2}$

Step2: Simplify the radical

$\sqrt{18x^3y^2} = \sqrt{9 \cdot 2 \cdot x^2 \cdot x \cdot y^2} = 3xy\sqrt{2x}$

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Problem 3

Step1: Combine radicals into one

$\frac{1}{3} \cdot \sqrt{\frac{144x^5}{2x}} = \frac{1}{3}\sqrt{72x^4}$

Step2: Simplify the radical

$\sqrt{72x^4} = \sqrt{36 \cdot 2 \cdot x^4} = 6x^2\sqrt{2}$

Step3: Multiply by coefficient

$\frac{1}{3} \cdot 6x^2\sqrt{2} = 2x^2\sqrt{2}$

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Problem 4

Step1: Multiply radicands (4th roots)

$\sqrt[4]{8xy^3 \cdot 6x^3y} = \sqrt[4]{48x^4y^4}$

Step2: Simplify the 4th root

$\sqrt[4]{48x^4y^4} = xy\sqrt[4]{16 \cdot 3} = 2xy\sqrt[4]{3}$

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Problem 5

Step1: Multiply coefficients & radicands

$9y \cdot x^2 \cdot \sqrt{12x \cdot 10xy^3} = 9x^2y\sqrt{120x^2y^3}$

Step2: Simplify the radical

$\sqrt{120x^2y^3} = \sqrt{4 \cdot 30 \cdot x^2 \cdot y^2 \cdot y} = 2xy\sqrt{30y}$

Step3: Multiply remaining terms

$9x^2y \cdot 2xy\sqrt{30y} = 18x^3y^2\sqrt{30y}$

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Problem 6

Step1: Simplify fraction inside radical

$\sqrt[4]{\frac{80xy^9}{5xy}} = \sqrt[4]{16y^8}$

Step2: Simplify the 4th root

$\sqrt[4]{16y^8} = \sqrt[4]{16 \cdot (y^2)^4} = 2y^2$

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Problem 7

Step1: Combine radicals into one

$\frac{7}{2} \cdot \sqrt[3]{\frac{320x^{11}y^5}{5x^8y^5}} = \frac{7}{2}\sqrt[3]{64x^3}$

Step2: Simplify the 3rd root

$\sqrt[3]{64x^3} = 4x$

Step3: Multiply by coefficient

$\frac{7}{2} \cdot 4x = 14x$

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Problem 8

Step1: Multiply coefficients & radicands

$3 \cdot \sqrt[3]{3x^4 \cdot 9x^2} = 3\sqrt[3]{27x^6}$

Step2: Simplify the 3rd root

$\sqrt[3]{27x^6} = 3x^2$

Step3: Multiply remaining terms

$3 \cdot 3x^2 = 9x^2$

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Problem 9

Step1: Multiply coefficients & radicands

$-3 \cdot 5 \cdot \sqrt[3]{4x^2 \cdot 16x} = -15\sqrt[3]{64x^3}$

Step2: Simplify the 3rd root

$\sqrt[3]{64x^3} = 4x$

Step3: Multiply remaining terms

$-15 \cdot 4x = -60x$

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Problem 10

Step1: Combine radicals into one

$\frac{3}{4} \cdot \sqrt{\frac{192x^6y^4}{3x^3y}} = \frac{3}{4}\sqrt{64x^3y^3}$

Step2: Simplify the radical

$\sqrt{64x^3y^3} = \sqrt{64 \cdot x^2 \cdot x \cdot y^2 \cdot y} = 8xy\sqrt{xy}$

Step3: Multiply by coefficient

$\frac{3}{4} \cdot 8xy\sqrt{xy} = 6xy\sqrt{xy}$

Answer:

1. $\boldsymbol{18x^2y\sqrt{2x}}$
2. $\boldsymbol{3xy\sqrt{2x}}$
3. $\boldsymbol{2x^2\sqrt{2}}$
4. $\boldsymbol{2xy\sqrt[4]{3}}$
5. $\boldsymbol{18x^3y^2\sqrt{30y}}$
6. $\boldsymbol{2y^2}$
7. $\boldsymbol{14x}$
8. $\boldsymbol{9x^2}$
9. $\boldsymbol{-60x}$
10. $\boldsymbol{6xy\sqrt{xy}}$