Question

1. simplify the expression below.

$2\sqrt{128} + 4\sqrt{20} - 2\sqrt{50}$
a. $14\sqrt{7}$
b. $4\sqrt{5} + 6\sqrt{2}$
c. $8\sqrt{5} - 6\sqrt{2}$
d. $8\sqrt{5} + 6\sqrt{2}$

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.

$4\sqrt{3}(\sqrt{6} - \sqrt{2})^2$
a. $32\sqrt{3}$
b. $32\sqrt{3} - 48$
c. $160\sqrt{3} - 48$
d. $8\sqrt{3}$

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.

$\frac{-6\sqrt{32}}{3\sqrt{3}}$
a. $-4\sqrt{6}$
b. $-\frac{2\sqrt{6}}{3}$
c. $-8\sqrt{2}$
d. $-\frac{8\sqrt{6}}{3}$

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. write the expression below in exponential form.

$\sqrt3{15k^4}$
a. $(15k)^{\frac{3}{4}}$
b. $(15k)^{\frac{4}{3}}$
c. $15^{\frac{1}{3}} \cdot k^{\frac{4}{3}}$
d. $15^3 \cdot k^{\frac{3}{4}}$

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:

Step1: Simplify each radical term

$2\sqrt{128} = 2\sqrt{64\times2}=2\times8\sqrt{2}=16\sqrt{2}$
$4\sqrt{20}=4\sqrt{4\times5}=4\times2\sqrt{5}=8\sqrt{5}$
$-2\sqrt{50}=-2\sqrt{25\times2}=-2\times5\sqrt{2}=-10\sqrt{2}$

Step2: Combine like radical terms

$16\sqrt{2}-10\sqrt{2}+8\sqrt{5}=6\sqrt{2}+8\sqrt{5}$

Step1: Combine cube roots

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

Step2: Multiply inside the radical

$\sqrt[3]{-135v^8} = \sqrt[3]{-27v^6 \times 5v^2}$

Step3: Simplify the cube root

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

Step1: Expand the squared binomial

$(\sqrt{6}-\sqrt{2})^2 = (\sqrt{6})^2 - 2\sqrt{6}\sqrt{2} + (\sqrt{2})^2 = 6 - 2\sqrt{12} + 2 = 8 - 4\sqrt{3}$

Step2: Distribute the outer term

$4\sqrt{3}(8 - 4\sqrt{3}) = 4\sqrt{3}\times8 - 4\sqrt{3}\times4\sqrt{3}$

Step3: Calculate each product

$32\sqrt{3} - 16\times3 = 32\sqrt{3} - 48$

Step1: Combine fourth roots

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

Step2: Simplify the fraction inside

$\sqrt[4]{80a^{15}} = \sqrt[4]{16a^{12} \times 5a^3}$

Step3: Simplify the fourth root

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

Answer:

D. $8\sqrt{5} + 6\sqrt{2}$

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