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Algebra II : Radicals

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Example Question #8 : Non Square Radicals

Simplify: $-\sqrt{ 120}$

Possible Answers:

$-2\sqrt{30}$

$-6\sqrt{5}$

$\textup{No solution.}$

$-4\sqrt{15}$

$-2\sqrt{10}$

Correct answer:

$-2\sqrt{30}$

Explanation:

In order to simplify this radical, rewrite the radical using common factors.

$-\sqrt{ 120} = -\sqrt{2} \cdot \sqrt{2}\cdot \sqrt{5} \cdot \sqrt{6}$

Simplify the square roots.

$-2\cdot \sqrt{5} \cdot \sqrt{6}$

Multiply the terms inside the radical.

The answer is: $-2\sqrt{30}$

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Example Question #9 : Non Square Radicals

Simplify: $\sqrt{30} \cdot \sqrt{15}$

Possible Answers:

$2\sqrt{15}$

$3\sqrt{10}$

$5\sqrt2$

$15\sqrt2$

$\textup{The radicals cannot be simplified any further.}$

Correct answer:

$15\sqrt2$

Explanation:

Break down the two radicals by their factors.

$\sqrt{30} \cdot \sqrt{15} = (\sqrt{3}\times\sqrt{2}\times \sqrt{5})\cdot(\sqrt{3}\times \sqrt{5})$

A square root of a number that is multiplied by itself is equal to the number inside the radical.

$\sqrt{3}\cdot \sqrt{3}=3$

$\sqrt{5}\cdot \sqrt{5}=5$

Simplify the terms in the parentheses.

$(\sqrt{3}\times\sqrt{2}\times \sqrt{5})\cdot(\sqrt{3}\times \sqrt{5}) = 3\times 5 \times \sqrt2$

The answer is: $15\sqrt2$

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Example Question #10 : Non Square Radicals

Simplify, if possible: $3\sqrt{5}+6\sqrt{50}+ 9\sqrt{500}$

Possible Answers:

$123\sqrt5$

$30\sqrt5+93\sqrt2$

$123\sqrt2$

$30\sqrt2+93\sqrt5$

$\textup{The radical cannot be simplified any further.}$

Correct answer:

$30\sqrt2+93\sqrt5$

Explanation:

The first term is already simplified. The second and third term will need to be simplified.

Write the common factors of the second radical and simplify.

$6\sqrt{50} =6\sqrt{25\times 2} = 6\sqrt{25}\times \sqrt{2} = 6(5)\sqrt2 = 30\sqrt2$

Repeat the process for the third term.

$9\sqrt{500} = 9\sqrt{100\times 5} = 9\sqrt{100}\times \sqrt{5} = 9(10)\sqrt{5} = 90\sqrt5$

Rewrite the expression.

$3\sqrt{5}+6\sqrt{50}+ 9\sqrt{500}= 3\sqrt{5}+30\sqrt2+90\sqrt5$

Combine like-terms.

The answer is: $30\sqrt2+93\sqrt5$

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Example Question #11 : Non Square Radicals

What is the value of $\sqrt{288}+\sqrt{48}$ ?

Possible Answers:

$16\sqrt6$

$12\sqrt2+3\sqrt2$

$16\sqrt5$

$12\sqrt2+4\sqrt3$

$12\sqrt3+4\sqrt2$

Correct answer:

$12\sqrt2+4\sqrt3$

Explanation:

Simplify the first term by using common factors of a perfect square.

$\sqrt{288} = \sqrt{144\times 2} = \sqrt{144} \cdot \sqrt{2} = 12\sqrt2$

Simplify the second term also by common factors.

$\sqrt{48} =\sqrt{4\cdot 4\cdot 3} =\sqrt{4}\cdot \sqrt{4}\cdot \sqrt{3} = 4\sqrt3$

Combine the terms.

$\sqrt{288}+\sqrt{48}=12\sqrt2+4\sqrt3$

The coefficients cannot be combined since these are unlike terms.

The answer is: $12\sqrt2+4\sqrt3$

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Example Question #12 : Non Square Radicals

Simplify: $\frac{\sqrt{32}}{\sqrt{6}}$

Possible Answers:

$\frac{4\sqrt3}{3}$

$\frac{\sqrt3}{2}$

$\frac{\sqrt3}{3}$

$\frac{\sqrt3}{4}$

$\frac{2\sqrt3}{3}$

Correct answer:

$\frac{4\sqrt3}{3}$

Explanation:

To simplify this, multiply the top and bottom by the denominator.

$\frac{\sqrt{32}}{\sqrt{6}}\cdot\frac{\sqrt{6}}{\sqrt{6}} = \frac{4\sqrt2\cdot \sqrt6}{6} =\frac{4\sqrt{12}}{6} =\frac{4\cdot 2\sqrt{3}}{6}$

Reduce the fraction.

The answer is: $\frac{4\sqrt3}{3}$

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Example Question #31 : Radicals

Simplify: $\sqrt{927}$

Possible Answers:

$9\sqrt{17}$

$103\sqrt{3}$

$3\sqrt{103}$

$12\sqrt{7}$

$9\sqrt{103}$

Correct answer:

$3\sqrt{103}$

Explanation:

To simplify radicals, we need to find perfect squares to factor out. In this case, it's .

$\sqrt{927}=\sqrt{9}*\sqrt{103}=3\sqrt{103}$ .

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Example Question #11 : Non Square Radicals

Simplify: $\sqrt{1248}$

Possible Answers:

$78\sqrt{14}$

$16\sqrt{13}$

$12\sqrt{78}$

$4\sqrt{78}$

$32\sqrt{39}$

Correct answer:

$4\sqrt{78}$

Explanation:

To simplify radicals, we need to find perfect squares to factor out. In this case, it's .

$\sqrt{1248}=\sqrt{16}*\sqrt{78}=4\sqrt{78}$

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Example Question #33 : Radicals

Simplify: $\sqrt{x^7y^3}$

Possible Answers:

$x^6y\sqrt{xy}$

$x^3y\sqrt{xy}$

$xy^2\sqrt{x^3y}$

$x^3y\sqrt{x^3y}$

$xy^2\sqrt{x^3y^2}$

Correct answer:

$x^3y\sqrt{xy}$

Explanation:

To simplify radicals, we need to find perfect squares to factor out. In this case, it's (x^3)^2,y^2 .

$\sqrt{x^7y^3}=\sqrt{x^6}*\sqrt{y^2}*\sqrt{xy}=\sqrt{(x^3)^2}*y\sqrt{xy}=x^3y\sqrt{xy}$

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Example Question #34 : Radicals

Simplify: $\sqrt{128x^4y^5}$

Possible Answers:

$8x^2y^2\sqrt{2xy}$

$64xy\sqrt{2xy}$

$64xy\sqrt{2y}$

$8x^2y^2\sqrt{2y}$

$8x^2y^2\sqrt{2x}$

Correct answer:

$8x^2y^2\sqrt{2y}$

Explanation:

To simplify radicals, we need to find perfect squares to factor out. In this case, it's 64,(x^2)^2, (y^2)^2 .

$\sqrt{128x^4y^5}=\sqrt{64}*\sqrt{(x^2)^2}*\sqrt{(y^2)^2}*\sqrt{2y}=8x^2y^2\sqrt{2y}$

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Example Question #3931 : Algebra Ii

Simplify: $2\sqrt{60} \cdot 3\sqrt{10}$

Possible Answers:

$24\sqrt{150}$

$30\sqrt6$

120

180

$60\sqrt6$

Correct answer:

$60\sqrt6$

Explanation:

Multiply the radicals.

$\sqrt{60} \times \sqrt{10} = \sqrt{600}$

Simplify this by writing the factors using perfect squares.

$\sqrt{600} = \sqrt{100} \times \sqrt{6} = 10\sqrt6$

Multiply this with the integers.

$2\sqrt{60} \cdot 3\sqrt{10} = 2\cdot 3 \cdot 10\sqrt6 = 60\sqrt6$

The answer is: $60\sqrt6$

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Algebra II : Radicals

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #8 : Non Square Radicals

Example Question #9 : Non Square Radicals

Example Question #10 : Non Square Radicals

Example Question #11 : Non Square Radicals

Example Question #12 : Non Square Radicals

Example Question #31 : Radicals

Example Question #11 : Non Square Radicals

Example Question #33 : Radicals

Example Question #34 : Radicals

Example Question #3931 : Algebra Ii

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