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Algebra II : Non-Square Radicals

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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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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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Algebra II : Non-Square Radicals

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

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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Example Question #10 : Non Square Radicals

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