Understanding Fractional Exponents

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Math › Understanding Fractional Exponents

Questions 1 - 3

1

Simplify the expression:

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

Explanation

Remember that fraction exponents are the same as radicals.

$\small 16^{\frac{1}{2}}=\sqrt{16}=4$

$256^{\frac{3}{4}}=\sqrt[4]{256^3}=64$

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

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

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

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

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

2

Convert the exponent to radical notation.

$x^{\frac{3}{7}}$

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

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

$\small \frac{x^3}{x^7}$

$\small \frac{1}{x^4}$

Explanation

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

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

$x^{\frac{3}{7}}=\sqrt[7]{x^3}$

3

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

$\frac{1}{32}$

Explanation

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

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

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