High School Math : Understanding Fractional Exponents

Study concepts, example questions & explanations for High School Math

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Example Questions

Example Question #1 : Fractional Exponents

Convert the exponent to radical notation.

\(\displaystyle x^{\frac{3}{7}}\)

Possible Answers:

\(\displaystyle \small \small \sqrt[3]{x^7}\)

\(\displaystyle \small \frac{1}{x^4}\)

\(\displaystyle \small \frac{x^3}{x^7}\)

\(\displaystyle \small \small \sqrt[7]{x^3}\)

Correct answer:

\(\displaystyle \small \small \sqrt[7]{x^3}\)

Explanation:

Remember that exponents in the denominator refer to the root of the term, while exponents in the numerator can be treated normally.

\(\displaystyle x^{\frac{a}{b}}=\sqrt[b]{x^a}\)

\(\displaystyle x^{\frac{3}{7}}=\sqrt[7]{x^3}\)

Example Question #1 : Exponents

Which of the following is equivalent to \(\displaystyle 64^{\frac{1}{2}}\) ?

Possible Answers:

\(\displaystyle -32\)

\(\displaystyle 8\)

\(\displaystyle 16\)

\(\displaystyle 32\)

\(\displaystyle \frac{1}{32}\)

Correct answer:

\(\displaystyle 8\)

Explanation:

By definition, a number raised to the \(\displaystyle \frac{1}{2}\) power is the same as the square root of that number. 

Since the square root of 64 is 8, 8 is our solution. 

Example Question #1 : Fractional Exponents

Simplify the expression:

\(\displaystyle \small (16^{\frac{1}{2}})(256^{\frac{3}{4}})\)

Possible Answers:

\(\displaystyle 256\)

\(\displaystyle 16\)

\(\displaystyle 64\)

\(\displaystyle 1024\)

Correct answer:

\(\displaystyle 256\)

Explanation:

Remember that fraction exponents are the same as radicals.

\(\displaystyle \small 16^{\frac{1}{2}}=\sqrt{16}=4\)

\(\displaystyle 256^{\frac{3}{4}}=\sqrt[4]{256^3}=64\)

A shortcut would be to express the terms as exponents and look for opportunities to cancel.

\(\displaystyle 16^{\frac{1}{2}}=(4^2)^{\frac{1}{2}}=4\)

\(\displaystyle \small 256^{\frac{3}{4}}=(4^4)^{\frac{3}{4}}=4^3=64\)

Either method, we then need to multiply to two terms.

\(\displaystyle \small (4)(64)=256\)

 

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