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High School Math : High School Math

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8 Diagnostic Tests 613 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #5 : Expressing Radicals As Exponents

Choose the fraction equivalent to $\frac{x^{\frac{1}{2}}-y^{\frac{1}{2}}}{x^{\frac{1}{2}}+y^{\frac{1}{2}}}$ .

Possible Answers:

$\frac{x+2x^{\frac{1}{2}}y^{\frac{1}{2}}+y}{x-y}$

$\frac{x-2x^{\frac{1}{2}}y^{\frac{1}{2}}+y}{x-y}$

$\frac{x+2x^{\frac{1}{2}}y^{\frac{1}{2}}+y}{x+y}$

$\frac{x-2x^{\frac{1}{2}}y^{\frac{1}{2}}+y}{x+y}$

Correct answer:

$\frac{x-2x^{\frac{1}{2}}y^{\frac{1}{2}}+y}{x-y}$

Explanation:

Multiply the numerator and denominator by the compliment of the denominator:

$=\frac{x^{\frac{1}{2}}-y^{\frac{1}{2}}}{x^{\frac{1}{2}}+y^{\frac{1}{2}}}$ $\cdot \frac{x^{\frac{1}{2}}-y^{\frac{1}{2}}}{x^{\frac{1}{2}}-y^{\frac{1}{2}}}$

Simplify the expression:

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

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Example Question #5 : Understanding Radicals

Simplify the following radical. Express in rational (exponential) form.

$(\frac{a^{\frac{-2}{3}}}{2^{\frac{1}{2}}a^2})^{\frac{-1}{2}}$

Possible Answers:

$2^{\frac{1}{3}}a^{\frac{4}{3}}$

$2^{\frac{1}{4}}a^{\frac{4}{3}}$

$2^{\frac{1}{4}}a^{\frac{5}{3}}$

$2^{\frac{1}{2}}a^{\frac{4}{3}}$

Correct answer:

$2^{\frac{1}{4}}a^{\frac{4}{3}}$

Explanation:

Multiply the numerator and denominator to the exponent:

$(\frac{a^{\frac{-2}{3}}}{2^{\frac{1}{2}}a^2})^{\frac{-1}{2}}$

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

Simplify the expression by combining like terms:

$=2^{\frac{1}{4}}a^{\frac{4}{3}}$

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Example Question #3 : Expressing Radicals As Exponents

Express the following radical in rational (exponential) form:

$\sqrt[3]{16x^5y^6z^8}$

Possible Answers:

$2^{\frac{4}{3}}x^{\frac{5}{3}}y^{\frac{7}{3}}z^{\frac{8}{3}}$

$2^{\frac{4}{3}}x^{\frac{5}{3}}y^2z^{\frac{8}{3}}$

$2^{\frac{4}{3}}x^2y^2z^{\frac{8}{3}}$

$2^2x^{\frac{5}{3}}y^2z^{\frac{8}{3}}$

$2^{\frac{4}{3}}x^{\frac{5}{3}}y^2z^2$

Correct answer:

$2^{\frac{4}{3}}x^{\frac{5}{3}}y^2z^{\frac{8}{3}}$

Explanation:

To convert the radical to exponent form, begin by converting the integer:

$\sqrt[3]{16x^5y^6z^8}$

$=\sqrt[3]{2^4x^5y^6z^8}$

Now, divide each exponent by to remove the radical:

$=2^{\frac{4}{3}}x^{\frac{5}{3}}y^{\frac{6}{3}}z^{\frac{8}{3}}$

Finally, simplify the exponents:

$=2^{\frac{4}{3}}x^{\frac{5}{3}}y^2z^{\frac{8}{3}}$

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Example Question #6 : Expressing Radicals As Exponents

Simplify: $(\frac{2^4a}{a^2b^{-1}})^{\frac{-1}{2}}$

Possible Answers:

$\frac{a^{\frac{1}{2}}b^{\frac{1}{2}}}{2b}$

$\frac{ab^{\frac{1}{2}}}{4b}$

$\frac{a^{\frac{1}{2}}}{4b^{\frac{1}{2}}}$

$\frac{a^{\frac{1}{2}}b}{4b}$

Correct answer:

$\frac{a^{\frac{1}{2}}}{4b^{\frac{1}{2}}}$

Explanation:

Multiply the numerator and denominator to the exponent:

$(\frac{2^4a}{a^2b^{-1}})^{\frac{-1}{2}}$

$=\frac{2^{-2}a^{\frac{-1}{2}}}{a^{-1}b^{\frac{1}{2}}}$

Simplify the expression by combining like terms:

$=\frac{a^{\frac{1}{2}}}{4b^{\frac{1}{2}}}$

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Example Question #51 : Algebra Ii

Express the following exponent in radical form:

$x^{\frac{1}{3}}y^{\frac{1}{2}}z^{\frac{3}{2}}$

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

Begin by converting each exponent to have a denominator of :

$x^{\frac{1}{3}}y^{\frac{1}{2}}z^{\frac{3}{2}}$

$=x^{\frac{2}{6}}y^{\frac{3}{6}}z^{\frac{9}{6}}$

Now, rearrange into radical form:

$=\sqrt[6]{x^2y^3z^9}$

Finally, simplify:

$=z\sqrt[6]{x^2y^3z^3}$

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Example Question #61 : Algebra Ii

Express the following exponent in radical form:

$r^2s^{\frac{1}{3}}y^{\frac{1}{2}}$

Possible Answers:

$r^2\sqrt[6]{s^2y^3}$

$r\sqrt[6]{s^2y^3}$

$r^6\sqrt[6]{s^2y^3}$

$r^4\sqrt[6]{s^2y^3}$

$r^3\sqrt[6]{s^2y^3}$

Correct answer:

$r^2\sqrt[6]{s^2y^3}$

Explanation:

Begin by converting each exponent to have a denominator of :

$r^2s^{\frac{1}{3}}y^{\frac{1}{2}}$

$=r^{\frac{12}{6}}s^{\frac{2}{6}}y^{\frac{3}{6}}$

Now, put this in radical form:

$=\sqrt[6]{r^{12}s^2y^3}$

Finally, simplify:

$=r^2\sqrt[6]{s^2y^3}$

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Example Question #62 : Algebra Ii

Simplify the following radical expression using exponents. Express the final answer in radical form.

$\sqrt[6]{8n^9}$

Possible Answers:

$n\sqrt{3n}$

$n\sqrt{2n}$

$2n\sqrt{2n}$

$2n\sqrt{3n}$

$3n\sqrt{3n}$

Correct answer:

$n\sqrt{2n}$

Explanation:

Begin by converting the radical into exponent form:

$\sqrt[6]{8n^9}$

$(8^{\frac{1}{6}}n^{\frac{9}{6}})$

Simplify the exponent and multiply:

$(2^3^{\frac{1}{6}}n^{\frac{9}{6}})$

$(2^{\frac{1}{2}}n^{\frac{3}{2}})$

Convert into radical form:

$\sqrt{2n^3}$

Simplify:

$n\sqrt{2n}$

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Example Question #11 : Understanding Radicals

Express the following exponent in radical form:

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

Possible Answers:

$\sqrt[4]{9x^3}$

$\sqrt[4]{7x^3}$

$\sqrt[4]{9x^2}$

$\sqrt[2]{9x^3}$

$\sqrt[2]{7x^3}$

Correct answer:

$\sqrt[4]{9x^3}$

Explanation:

Begin by changing the fractional exponents so that they both have a common denominator of :

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

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

Now, put this in radical form and simplify:

$=\sqrt[4]{(3x)^2x}$

$=\sqrt[4]{9x^2x}$

$=\sqrt[4]{9x^3}$

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Example Question #1 : Simplifying Radicals

Factor and simplify the following radical expression:

$\sqrt[4]{81x^6y^5z^{10}}$

Possible Answers:

$3xyz^2\sqrt[4]{x^2y^2z^2}$

$3xy^2z^2\sqrt[4]{x^2yz^2}$

$3xyz\sqrt[4]{x^2yz^2}$

$3x^2yz^2\sqrt[4]{x^2yz^2}$

$3xyz^2\sqrt[4]{x^2yz^2}$

Correct answer:

$3xyz^2\sqrt[4]{x^2yz^2}$

Explanation:

Begin by factoring the integer:

$\sqrt[4]{81x^6y^5z^{10}}$

$\sqrt[4]{3^4x^6y^5z^{10}}$

$3\sqrt[4]{x^6y^5z^{10}}$

Now, simplify the exponents:

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

$3xyzz\sqrt[4]{x^2yz^2}$

$3xyz^2\sqrt[4]{x^2yz^2}$

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Example Question #2 : Simplifying Radicals

Factor and simplify the following radical expression:

$\sqrt[3]{(3n-2)^3}$

Possible Answers:

n-2

(3n-2)^2

3n+2

3n-2

(3n-2)^3

Correct answer:

3n-2

Explanation:

Begin by converting the radical into exponent form:

$\sqrt[3]{(3n-2)^3}$

$(3n-2)^3^{(\frac{1}{3})}$

Now, multiply the exponents:

3n-2

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High School Math : High School Math

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #5 : Expressing Radicals As Exponents

Example Question #5 : Understanding Radicals

Example Question #3 : Expressing Radicals As Exponents

Example Question #6 : Expressing Radicals As Exponents

Example Question #51 : Algebra Ii

Example Question #61 : Algebra Ii

Example Question #62 : Algebra Ii

Example Question #11 : Understanding Radicals

Example Question #1 : Simplifying Radicals

Example Question #2 : Simplifying Radicals

All High School Math Resources

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