Question

#1,3,5,7
name
unit 6: radical functions
date: __________ per: ______ homework 3: add/subtract/multiply radicals
directions: simplify each expression below.

1. $7\sqrt3{3} + 2\sqrt3{192}$
2. $10\sqrt{7} - \sqrt{28} + 6\sqrt{180}$
3. $8\sqrt4{48} - 5\sqrt{90} + 9\sqrt4{3}$
4. $5\sqrt3{32x^{3}y^{4}} - 3xy\sqrt3{4y}$
5. $15\sqrt{2} \cdot -2\sqrt{20}$
6. $4\sqrt3{-9} \cdot 7\sqrt3{48}$
7. $3\sqrt{18w^{7}} \cdot 10\sqrt{4w^{9}}$
8. $\sqrt4{8a^{3}b} \cdot \sqrt4{10a^{2}b^{7}}$

Explanation:

Step1: Simplify $\sqrt[3]{192}$

$\sqrt[3]{192} = \sqrt[3]{64 \times 3} = 4\sqrt[3]{3}$

Step2: Substitute back and combine terms

$7\sqrt[3]{3} + 2 \times 4\sqrt[3]{3} = 7\sqrt[3]{3} + 8\sqrt[3]{3} = 15\sqrt[3]{3}$

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Step1: Simplify $\sqrt{28}$ and $\sqrt{180}$

$\sqrt{28} = 2\sqrt{7}$, $\sqrt{180} = 6\sqrt{5}$

Step2: Substitute back and combine terms

$10\sqrt{7} - 2\sqrt{7} + 6 \times 6\sqrt{5} = 8\sqrt{7} + 36\sqrt{5}$

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Step1: Simplify $\sqrt[4]{48}$

$\sqrt[4]{48} = \sqrt[4]{16 \times 3} = 2\sqrt[4]{3}$

Step2: Simplify $\sqrt{90}$

$\sqrt{90} = 3\sqrt{10}$

Step3: Substitute back and combine terms

$8 \times 2\sqrt[4]{3} - 5 \times 3\sqrt{10} + 9\sqrt[4]{3} = 16\sqrt[4]{3} + 9\sqrt[4]{3} - 15\sqrt{10} = 25\sqrt[4]{3} - 15\sqrt{10}$

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Step1: Simplify $\sqrt[3]{32x^3y^4}$

$\sqrt[3]{32x^3y^4} = \sqrt[3]{8x^3y^3 \times 4y} = 2xy\sqrt[3]{4y}$

Step2: Substitute back and combine terms

$5 \times 2xy\sqrt[3]{4y} - 3xy\sqrt[3]{4y} = 10xy\sqrt[3]{4y} - 3xy\sqrt[3]{4y} = 7xy\sqrt[3]{4y}$

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Step1: Simplify $\sqrt{20}$

$\sqrt{20} = 2\sqrt{5}$

Step2: Substitute back

$15\sqrt{2} - 2 \times 2\sqrt{5} = 15\sqrt{2} - 4\sqrt{5}$

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Step1: Simplify $\sqrt[3]{48}$

$\sqrt[3]{48} = \sqrt[3]{8 \times 6} = 2\sqrt[3]{6}$

Step2: Multiply the terms

$4\sqrt[3]{-9} \times 7 \times 2\sqrt[3]{6} = 56\sqrt[3]{-9 \times 6} = 56\sqrt[3]{-54}$

Step3: Simplify $\sqrt[3]{-54}$

$\sqrt[3]{-54} = \sqrt[3]{-27 \times 2} = -3\sqrt[3]{2}$

Step4: Final calculation

$56 \times (-3\sqrt[3]{2}) = -168\sqrt[3]{2}$

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Step1: Simplify $\sqrt{18w^7}$ and $\sqrt{4w^9}$

$\sqrt{18w^7} = 3w^3\sqrt{2w}$, $\sqrt{4w^9} = 2w^4\sqrt{w}$

Step2: Substitute back

$3 \times 3w^3\sqrt{2w} - 10 \times 2w^4\sqrt{w} = 9w^3\sqrt{2w} - 20w^4\sqrt{w}$

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Step1: Combine the 4th roots

$\sqrt[4]{8a^3b} \cdot \sqrt[4]{10a^2b^7} = \sqrt[4]{8a^3b \times 10a^2b^7}$

Step2: Multiply inside the root

$\sqrt[4]{80a^5b^8} = \sqrt[4]{16a^4b^8 \times 5a} = 2ab^2\sqrt[4]{5a}$

Answer:

1. $\boldsymbol{15\sqrt[3]{3}}$
2. $\boldsymbol{8\sqrt{7} + 36\sqrt{5}}$
3. $\boldsymbol{25\sqrt[4]{3} - 15\sqrt{10}}$
4. $\boldsymbol{7xy\sqrt[3]{4y}}$
5. $\boldsymbol{15\sqrt{2} - 4\sqrt{5}}$
6. $\boldsymbol{-168\sqrt[3]{2}}$
7. $\boldsymbol{9w^3\sqrt{2w} - 20w^4\sqrt{w}}$
8. $\boldsymbol{2ab^2\sqrt[4]{5a}}$