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Algebra II : Radicals

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Example Question #21 : Square Roots

Evaluate: $\sqrt{0}-\sqrt1-\sqrt9-\sqrt{16}$

Possible Answers:

$\textup{Undefined.}$

Correct answer:

Explanation:

Evaluate each square root. The square root of a number evaluates into a number which multiplies by itself to achieve the number in the square root.

$\sqrt0 =0$

$\sqrt{1} = 1$

$\sqrt{9} =3$

$\sqrt{16}=4$

Substitute the terms back into the expression.

$\sqrt{0}-\sqrt1-\sqrt9-\sqrt{16} = 0-1-3-4 = -8$

The answer is:

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Example Question #22 : Square Roots

Solve: $\sqrt{16}+\sqrt{196}-\sqrt{36}$

Possible Answers:

Correct answer:

Explanation:

Solve each radical. The square root determines a number that multiples by itself to equal the number inside the square root.

$\sqrt{16}=4$

$\sqrt{196}=14$

$\sqrt{36}=6$

Rewrite the expression.

$\sqrt{16}+\sqrt{196}-\sqrt{36} = 4+14-6$

The answer is:

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Example Question #21 : Radicals

True or false: $\sqrt{52}$ is a radical expression in simplest form.

Possible Answers:

True

False

Correct answer:

False

Explanation:

A radical expression which is the th root of a constant is in simplest form if and only if, when the radicand is expressed as the product of prime factors, no factor appears or more times. Since $\sqrt{52}$ is a square, or second, root, find the prime factorization of 52, and determine whether any prime factor appears two or more times.

52 can be broken down as

$52 = 2 \times 26$

and further as

$52 = 2 \times 2 \times 13$

The factor 2 appears twice, so $\sqrt{52}$ is not in simplest form.

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Example Question #1 : Non Square Radicals

$(x^{2})^{\frac{1}{3}}(x^{-2})^{\frac{1}{2}}(x^{4})^{\frac{1}{3}} =$

Possible Answers:

$x^{\frac{-8}{9}}$

$x^{\frac{4}{3}}$

$x^{3}$

Correct answer:

Explanation:

To solve this, remember that when multiplying variables, exponents are added. When raising a power to a power, exponents are multiplied. Thus:

$(x^{2})^{\frac{1}{3}}(x^{-2})^{\frac{1}{2}}(x^{4})^{\frac{1}{3}} =$

$x^{\frac{2}{3}}x^{-1}x^{\frac{4}{3}} = x^{\frac{2}{3}-1+\frac{4}{3}} = x$

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Example Question #2 : Non Square Radicals

Simplify by rationalizing the denominator:

$\frac{6}{\sqrt[3]{18} }$

Possible Answers:

$6 \sqrt[3]{12}$

$2 \sqrt[3]{2}$

$\sqrt[3]{2}$

$6 \sqrt[3]{2}$

$\sqrt[3]{12}$

Correct answer:

$\sqrt[3]{12}$

Explanation:

Since $18 = 2 \cdot 3 ^{2}$ , we can multiply 18 by $2^{2} \cdot 3 = 12$ to yield the lowest possible perfect cube:

$18 \cdot 12 = 216 = 6 ^{3}$

Therefore, to rationalize the denominator, we multiply both nuerator and denominator by $\sqrt[3]{12}$ as follows:

$\frac{6}{\sqrt[3]{18} }$

$= \frac{6 \cdot \sqrt[3]{12} }{\sqrt[3]{18} \cdot \sqrt[3]{12} }$

$= \frac{6 \cdot \sqrt[3]{12} }{\sqrt[3]{18 \cdot 12 } }$

$= \frac{6 \sqrt[3]{12} }{\sqrt[3]{216 } }$

$= \frac{6 \sqrt[3]{12} }{6 }$

$= \sqrt[3]{12}$

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Example Question #3 : Non Square Radicals

Simplify: $\sqrt[3]{120000}$

Possible Answers:

$120\sqrt[3]{5}$

$25\sqrt[3]{4}$

240

$20\sqrt[3]{15}$

$10\sqrt[3]{120}$

Correct answer:

$20\sqrt[3]{15}$

Explanation:

Begin by getting a prime factor form of the contents of your root.

120000 = 12 * 10000 = 2^2*3*10^4

Applying some exponent rules makes this even faster:

10^4 = (2*5)^4=2^45^4

Put this back into your problem:

2^2*3*2^4*5^4=2^6*3*5^4

Returning to your radical, this gives us:

$\sqrt[3]{2^6*3*5^4}$

Now, we can factor out sets of 2^3 and set of 5^3 . This gives us:

$2^2*5\sqrt[3]{3*5}=20\sqrt[3]{15}$

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Example Question #4 : Non Square Radicals

Simplify:

$\sqrt[4]{15625}$

Possible Answers:

$125\sqrt[4]{5}$

$5\sqrt{35}$

$5\sqrt[4]{5}$

$125\sqrt{5}$

$5\sqrt{5}$

Correct answer:

$5\sqrt{5}$

Explanation:

Begin by factoring the contents of the radical:

15625 = 25*625 = 25*25*25=5*5*5*5*5*5=5^6

This gives you:

$\sqrt[4]{5^6}$

You can take out group of 5^4 . That gives you:

$5\sqrt[4]{5^2}=5\sqrt[4]{25}$

Using fractional exponents, we can rewrite this:

$5*5^\frac{2}{4}$

Thus, we can reduce it to:

$5*5^\frac{1}{2}$

Or:

$5\sqrt{5}$

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Example Question #5 : Non Square Radicals

Simplify: $\sqrt{125}+\sqrt{50}$

Possible Answers:

$15\sqrt{10}$

$5\sqrt7$

$5\sqrt5+5\sqrt2$

$10\sqrt5+10\sqrt2$

$10\sqrt{10}$

Correct answer:

$5\sqrt5+5\sqrt2$

Explanation:

To simplify $\sqrt{125}+\sqrt{50}$ , find the common factors of both radicals.

$\sqrt{125}= \sqrt{25\cdot5}=\sqrt{25}\cdot \sqrt5=5\sqrt5$

$\sqrt{50}=\sqrt{25\cdot 2}=\sqrt{25}\cdot \sqrt2=5\sqrt2$

Sum the two radicals.

The answer is: $5\sqrt5+5\sqrt2$

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Example Question #6 : Non Square Radicals

Simplify:

$\sqrt[3]{27x^5y^6}$

Possible Answers:

$xy^2\sqrt[3]{x^2}$

$3xy^2\sqrt[3]{x^2}$

$3xy\sqrt[3]{x^2}$

$3xy^2\sqrt[3]{x}$

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

Correct answer:

$3xy^2\sqrt[3]{x^2}$

Explanation:

To take the cube root of the term on the inside of the radical, it is best to start by factoring the inside:

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

Now, we can identify three terms on the inside that are cubes:

x^3, y^6, 27

We simply take the cube root of these terms and bring them outside of the radical, leaving what cannot be cubed on the inside of the radical.

$x\cdot y^2 \cdot 3 \cdot \sqrt[3]{x^2}$

Rewritten, this becomes

$3xy^2\sqrt[3]{x^2}$

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Example Question #7 : Non Square Radicals

Simplify the radical: $\sqrt{32}\cdot\sqrt{8}$

Possible Answers:

$4\sqrt2$

$8\sqrt2$

Correct answer:

Explanation:

Simplify both radicals by rewriting each of them using common factors.

$\sqrt{32} = \sqrt4 \cdot \sqrt{4}\cdot \sqrt{2} = 4\sqrt2$

$\sqrt{8} = \sqrt4 \cdot \sqrt2 = 2\sqrt2$

Multiply the two radicals.

$4\sqrt2\cdot 2\sqrt2 = 4\cdot 2 \cdot 2 = 16$

The answer is:

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Algebra II : Radicals

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #21 : Square Roots

Example Question #22 : Square Roots

Example Question #21 : Radicals

Example Question #1 : Non Square Radicals

Example Question #2 : Non Square Radicals

Example Question #3 : Non Square Radicals

Example Question #4 : Non Square Radicals

Example Question #5 : Non Square Radicals

Example Question #6 : Non Square Radicals

Example Question #7 : Non Square Radicals

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