Expressing Radicals as Exponents

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Convert the radical to exponential notation.

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

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

CORRECT

$\small \sqrt{13^4}$

$\small 169$

$\small 13^4$

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.

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

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

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

CORRECT

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

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

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

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:

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

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

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

CORRECT

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

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

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

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:

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

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

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

CORRECT

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

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

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

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:

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

Express the following exponent in radical form:

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

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

CORRECT

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

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

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

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

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:

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

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

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

$n\sqrt{2n}$

CORRECT

$n\sqrt{3n}$

$2n\sqrt{2n}$

$2n\sqrt{3n}$

$3n\sqrt{3n}$

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:

$n\sqrt{2n}$

Express the following exponent in radical form:

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

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

CORRECT

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

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

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

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

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:

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

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

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

Express the following radical in rational (exponential) form:

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

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

CORRECT

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

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

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

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:

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

Express the following radical in rational (exponential) form:

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

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

CORRECT

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

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

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

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

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:

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

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

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

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

CORRECT

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

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

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

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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