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Algebra II : Radicals

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Example Questions

Example Question #28 : Simplifying Radicals

Simplify: $\sqrt{96}$

Possible Answers:

None of these

$4\sqrt{6}$

$6\sqrt{2}$

$2\sqrt{12}$

$2\sqrt{6}$

Correct answer:

$4\sqrt{6}$

Explanation:

To simplify a radical that is not a perfect square (such as 25 or 9) we must factor the radicand to numbers that are perfect squares. Try and break up the number to factors that you can take the perfect square of.

$\sqrt{96}$

$\sqrt{12*8}$

$\sqrt{2*6*4*2}$

Extract perfect squares:

$2*2\sqrt{6}$

$\mathbf{4\sqrt{6}}$

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Example Question #29 : Simplifying Radicals

Simplify: $\sqrt{48x^9y^{12}}$

Possible Answers:

4x^4y^6

$4x^4y^6\sqrt{3x}$

$4x^2y^6\sqrt{3x}$

$2x^4y^6\sqrt{3x}$

$4x^4y^6\sqrt{x}$

Correct answer:

$4x^4y^6\sqrt{3x}$

Explanation:

To simplify this radical, work on each term separately. $\sqrt{48}=\sqrt{3\cdot 2\cdot 2\cdot 2\cdot 2}=4\sqrt{3}$ , $\sqrt{x^9}=x^4\sqrt{x}$ , and $\sqrt{y^{12}}=y^6$ . Put those all together to get your answer of $4x^4y^6\sqrt{3x}$ .

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Example Question #121 : Radicals

Simplify the radical to its simplest form.

$\sqrt[3]{128}$

Possible Answers:

$3{\sqrt[3]{16}}$

$4\sqrt[3]{2}$

$2\sqrt[3]{4}$

Correct answer:

$4\sqrt[3]{2}$

Explanation:

In order to simplify the radical, you need to think of all of the factors of the radicand (the number under the radical sign). The factors of 128 are (1,2,4,8,16,32,64,128) . The greatest cube root of 128 is 64. $\sqrt[3]{64\times 2}$ . You can take the 64 out, because the cube root of 64 is 4. $4\sqrt[3]{2}$ is what is left.

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Example Question #121 : Radicals

Simplify $\sqrt{300}$

Possible Answers:

$5\sqrt{12}$

$10\sqrt{30}$

$10\sqrt{3}$

$100\sqrt{3}$

Cannot be simplified further.

Correct answer:

$10\sqrt{3}$

Explanation:

To simplify a square root, factor out perfect squares:

$\sqrt{300}=\sqrt{100}*\sqrt{3}=10\sqrt{3}$

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Example Question #32 : Simplifying Radicals

Simplify $\sqrt{512x^{5}}$ .

Possible Answers:

$x^{2}\sqrt{512x}$

$16\sqrt{2x^{5}}$

$16x^{2}\sqrt{2x}$

$256x^{2}\sqrt{2x}$

$16x\sqrt{2x}$

Correct answer:

$16x^{2}\sqrt{2x}$

Explanation:

While neither 512 or $x^{5}$ is a perfect square, they both contain factors that are perfect squares that can be taken out from under the radical.

Remember the Product of a Power property of exponents can also apply to radicals, so:

$\sqrt{ab}=\sqrt{a}\cdot \sqrt{b}$

Using this property, you get:

$\sqrt{512x^{5}}=\sqrt{512}\cdot \sqrt{x^{5}}$

Both of these can be factored. Let's look at each term separately, and after we've factored them, we can recombine the terms at the end.

First look for a factor of 512 that is a perfect square. One factor of 512 is 256, which has a square root of 16, so:

$\sqrt{512}=\sqrt{256}\cdot \sqrt{2}=\sqrt{16^{2}}\cdot \sqrt{2}=16\sqrt{2}$

Next look for a factor of $x^{5}$ that is a perfect square.

$\sqrt{x^{5}}=\sqrt{x^{4}}\cdot \sqrt{x}=\sqrt{(x^{2})^{2}}\cdot \sqrt{x}=x^{2}\sqrt{x}$

Now combine these two terms:

$(16\sqrt{2})(x^{2}\sqrt{x})=16x^{2}\sqrt{2x}$

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Example Question #122 : Radicals

Simplify $6\sqrt{24x^{2}}$ .

Possible Answers:

$12x\sqrt{6}$

$6x^{2}\sqrt{24}$

$24x\sqrt{6}$

$8x\sqrt{6}$

$6x\sqrt{24}$

Correct answer:

$12x\sqrt{6}$

Explanation:

While 24 is not a perfect square it does contain a factor that is a perfect square. Use the Power of a Product property to rewrite the problem:

$6\sqrt{24x^{2}}=6\cdot \sqrt{24}\cdot \sqrt{x^{2}}$

24 contains the factors 4 and 6.

$6\cdot \sqrt{24}\cdot \sqrt{x^{2}}=6\cdot \sqrt{4}\cdot \sqrt{6}\cdot \sqrt{x^{2}}=6\cdot \sqrt{2^{2}}\cdot \sqrt{6}\cdot \sqrt{x^{2}}$

$=6\cdot 2\cdot \sqrt{6}\cdot x=12x\sqrt{6}$

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Example Question #4022 : Algebra Ii

Simplify:

$\sqrt{84x^6y^7z^9}$

Possible Answers:

$2x^3y^3z^4\sqrt{21yz}$

$x^3y^3z^4\sqrt{21yz}$

$2xy^3z^4\sqrt{21yz}$

$2x^3y^3z^4\sqrt{2yz}$

$2x^3y^4z^4\sqrt{21yz}$

Correct answer:

$2x^3y^3z^4\sqrt{21yz}$

Explanation:

To simplify this radical, focus on each term separately. Recall that with square roots, for every pair of the same number or term, you can cross that under the radical and put one of them outside it. I'd recommend factoring 84 to its prime factors:

$84=2\cdot 2\cdot 3\cdot 7$ .

Therefore, you can take a 2 outside of the radical and leave 21 inside.

$\sqrt{x^6}=x^3$ ,

$\sqrt{y^7}=y^3\sqrt{y}$ ,

and

$\sqrt{z^9}=z^4\sqrt{z}$ .

Put these all together to get your final answer of

$2x^3y^3z^4\sqrt{21yz}$ .

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Example Question #35 : Simplifying Radicals

Factor the radical: $\sqrt{88}$

Possible Answers:

$4\sqrt{22}$

$4\sqrt{11}$

$8\sqrt{11}$

$2\sqrt{11}$

$2\sqrt{22}$

Correct answer:

$2\sqrt{22}$

Explanation:

The radical can be rewritten with common factors.

Pull out the factor of a known square.

$\sqrt{88} =\sqrt{4\times 22} = \sqrt4 \cdot \sqrt{22} = 2\sqrt{22}$

The value of $\sqrt{22}$ cannot be broken down any further.

The answer is: $2\sqrt{22}$

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Example Question #36 : Simplifying Radicals

Simplify the radicals, if possible: $\sqrt{30}\times \sqrt{60}$

Possible Answers:

$6\sqrt{30}$

$2\sqrt{30}$

$30\sqrt2$

$3\sqrt{30}$

$12\sqrt{30}$

Correct answer:

$30\sqrt2$

Explanation:

Simplify each term.

$\sqrt{30} =\sqrt{2}\cdot \sqrt{3}\cdot \sqrt{5}$

$\sqrt{60} =\sqrt{2} \cdot \sqrt{2}\cdot \sqrt{3}\cdot \sqrt{5}$

Multiply these terms together. Identical radicals multiplied by each other will cancel out the radical and leave the integer.

$(\sqrt{2}\cdot \sqrt{3}\cdot \sqrt{5})(\sqrt{2}\cdot\sqrt{2}\cdot \sqrt{3}\cdot \sqrt{5})$

$2\cdot3\cdot5\cdot \sqrt2 = 30\sqrt2$

The answer is: $30\sqrt2$

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Example Question #37 : Simplifying Radicals

Simplify: $\sqrt{24}+\sqrt{54}$

Possible Answers:

$6\sqrt{2}$

$2\sqrt{19}$

$6\sqrt5$

$5\sqrt6$

$3\sqrt5$

Correct answer:

$5\sqrt6$

Explanation:

To add the radicals, first factor out the terms using known perfect squares.

$\sqrt{24}+\sqrt{54} = \sqrt{4}\cdot \sqrt{6}+ \sqrt{9}\cdot \sqrt{6}$

Simplify the radicals with perfect squares.

$\sqrt{24}+\sqrt{54} = 2\cdot \sqrt{6}+3\cdot \sqrt{6}$

Add the like terms.

The answer is: $5\sqrt6$

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Algebra II : Radicals

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #28 : Simplifying Radicals

Example Question #29 : Simplifying Radicals

Example Question #121 : Radicals

Example Question #121 : Radicals

Example Question #32 : Simplifying Radicals

Example Question #122 : Radicals

Example Question #4022 : Algebra Ii

Example Question #35 : Simplifying Radicals

Example Question #36 : Simplifying Radicals

Example Question #37 : Simplifying Radicals

All Algebra II Resources

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