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High School Math : Understanding Radicals

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Example Question #1 : Understanding Square Roots

Simplify the expression. Find the positive solution only.

$\sqrt{50x^3y^4}$

Possible Answers:

$25x^2y^4\sqrt{2x}$

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

$xy^2\sqrt{50xy^2}$

25xy

$5xy\sqrt{2x}$

Correct answer:

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

Explanation:

When working in square roots, each component can be treated separately.

$\sqrt{50x^3y^4}=\sqrt{50}\sqrt{x^3}\sqrt{y^4}$

Now, we can simplify each term.

$\sqrt{50}=5\sqrt{2}$

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

$\sqrt{y^4}=y^2$

Combine the simplified terms to find the answer. Anything outside of the square root is combined, while anything under the root is combined under the root.

$(5\sqrt{2})(x\sqrt{x})(y^2)=5xy^2\sqrt{2x}$

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Example Question #1 : Understanding Radicals

Convert the radical to exponential notation.

$\small \sqrt[4]{13}$

Possible Answers:

$13^{\frac{1}{4}}$

$\small 13^4$

$\small 169$

$\small \sqrt{13^4}$

Correct answer:

$13^{\frac{1}{4}}$

Explanation:

Remember that any term outside the radical will be in the denominator of the exponent.

$\sqrt[b]{x^a}=x^{\frac{a}{b}}$

Since does not have any roots, we are simply raising it to the one-fourth power.

$\sqrt[4]{13}=13^{\frac{1}{4}}$

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Example Question #2 : Expressing Radicals As Exponents

What is the value of $9^ \frac{5}{2}$ ?

Possible Answers:

243

2.41

Correct answer:

243

Explanation:

An exponent written as a fraction can be rewritten using roots. $9^ \frac{5}{2}$ can be reqritten as $\sqrt[2]{9^5}$ . The bottom number on the fraction becomes the root, and the top becomes the exponent you raise the number to. $\sqrt[2]{9^5}$ is the same as $(\sqrt[2]{9})^{5}=3^{5}$ . This will give us the answer of 243.

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Example Question #3 : Expressing Radicals As Exponents

Express the following radical in rational (exponential) form:

$\sqrt{8x^3y^4}$

Possible Answers:

$2^2x^{(\frac{3}{2})}y^2$

2^2x^2y^2

$2^{(\frac{3}{2})}x^2y^2$

$2^{(\frac{3}{2})}x^{(\frac{3}{2})}y^{(\frac{3}{2})}$

$2^{(\frac{3}{2})}x^{(\frac{3}{2})}y^2$

Correct answer:

$2^{(\frac{3}{2})}x^{(\frac{3}{2})}y^2$

Explanation:

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

$\sqrt{8x^3y^4}$

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

Now, divide each exponent by to clear the square root:

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

Finally, simplify the exponents:

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

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Example Question #4 : Expressing Radicals As Exponents

Express the following radical in rational (exponential) form:

$\sqrt[4]{96a^5b^7c^8}$

Possible Answers:

$2^{\frac{5}{4}}a^{\frac{5}{4}}b^{\frac{7}{4}}c^2$

$3^{\frac{1}{4}}2^{\frac{5}{4}}a^{\frac{1}{4}}b^{\frac{7}{4}}c^2$

$3^{\frac{1}{4}}2^{\frac{5}{4}}a^{\frac{5}{4}}b^{\frac{7}{4}}$

$3^{\frac{5}{4}}a^{\frac{5}{4}}b^{\frac{7}{4}}c^2$

$3^{\frac{1}{4}}2^{\frac{5}{4}}a^{\frac{5}{4}}b^{\frac{7}{4}}c^2$

Correct answer:

$3^{\frac{1}{4}}2^{\frac{5}{4}}a^{\frac{5}{4}}b^{\frac{7}{4}}c^2$

Explanation:

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

$\sqrt[4]{96a^5b^7c^8}$

$=\sqrt[4]{3\cdot 2^5a^5b^7c^8}$

Now, divide each exponent by to cancel the radical:

$=3^{\frac{1}{4}}2^{\frac{5}{4}}a^{\frac{5}{4}}b^{\frac{7}{4}}c^{\frac{8}{4}}$

Finally, simplify the exponents:

$=3^{\frac{1}{4}}2^{\frac{5}{4}}a^{\frac{5}{4}}b^{\frac{7}{4}}c^2$

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Example Question #5 : Expressing Radicals As Exponents

Which fraction is equivalent to $\frac{x^{\frac{5}{2}}}{x^{\frac{1}{2}}+y^{\frac{1}{2}}}$ ?

Possible Answers:

$\frac{x^3+x^{\frac{5}{2}}y^\frac{1}{2}}{x+y}$

$\frac{x^3+x^{\frac{5}{2}}y^\frac{1}{2}}{x-y}$

$\frac{x^3-x^{\frac{5}{2}}y^\frac{1}{2}}{x-y}$

$\frac{x^3-x^{\frac{5}{2}}y^\frac{1}{2}}{x+y}$

Correct answer:

$\frac{x^3-x^{\frac{5}{2}}y^\frac{1}{2}}{x-y}$

Explanation:

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

$=\frac{x^{\frac{5}{2}}}{x^{\frac{1}{2}}+y^{\frac{1}{2}}} \cdot \frac{x^{\frac{1}{2}}-y^{\frac{1}{2}}}{x^{\frac{1}{2}}-y^{\frac{1}{2}}}$

Simplify the expression:

$=\frac{x^3-x^{\frac{5}{2}}y^\frac{1}{2}}{x-y}$

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Example Question #6 : Expressing Radicals As Exponents

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

$\frac{x^{\frac{2}{3}}}{x^{\frac{2}{3}}-x^{\frac{-1}{3}}}$

Possible Answers:

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

$\frac{x}{x-1}$

$\frac{x}{x-2}$

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

Correct answer:

$\frac{x}{x-1}$

Explanation:

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

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

Simplify the expression:

$=\frac{x}{x-1}$

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Example Question #7 : 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 #8 : Expressing Radicals As Exponents

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:

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

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

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

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

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 #9 : Expressing Radicals As Exponents

Express the following radical in rational (exponential) form:

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

Possible Answers:

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

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

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

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

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

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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High School Math : Understanding Radicals

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #1 : Understanding Square Roots

Example Question #1 : Understanding Radicals

Example Question #2 : Expressing Radicals As Exponents

Example Question #3 : Expressing Radicals As Exponents

Example Question #4 : Expressing Radicals As Exponents

Example Question #5 : Expressing Radicals As Exponents

Example Question #6 : Expressing Radicals As Exponents

Example Question #7 : Expressing Radicals As Exponents

Example Question #8 : Expressing Radicals As Exponents

Example Question #9 : Expressing Radicals As Exponents

All High School Math Resources

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