Algebra II - Fractional Exponents

Evaluate $17^\frac{1}{2}$

$\sqrt{17}$

$\sqrt{8.5}$

Explanation

When dealing with fractional exponents, we rewrite as such:

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

in which is the index of the radical and is the exponent raising base .

$17^\frac{1}{2}=\sqrt{17}$

Evaluate $2^{\frac{1}2{}}$

$\sqrt{2}$

$\frac{1}{4}$

$\frac{\sqrt{2}}2{}$

Explanation

When dealing with fractional exponents, remember this form:

$\sqrt[a]{x^n}$ is the index of the radical which is also the denominator of the fraction, represents the base of the exponent, and is the power the base is raised to. That value is the numerator of the fraction.

$2^{\frac{1}{2}}=\sqrt[2]{2^1}=\sqrt{2}$

Evaluate $3^{\frac{1}{3}}$

$\sqrt[3]{3}$

$\sqrt{3}$

$\frac{1}{27}$

$\frac{\sqrt[3]{3}}{3}$

Explanation

When dealing with fractional exponents, remember this form:

$3^{\frac{1}{3}}=\sqrt[3]{3^1}=\sqrt[3]{3}$

Evaluate $18^\frac{2}{3}$

$\sqrt[3]{324}$

$8\sqrt{2}$

$\sqrt[3]{54}$

$\sqrt{5832}$

Explanation

When dealing with fractional exponents, we rewrite as such:

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

in which is the index of the radical and is the exponent raising base .

$18^\frac{2}{3}=\sqrt[3]{18^2}=\sqrt[3]{324}$

Write $x^{2/5}$ in radical form.

$\sqrt[5]{x^{2}}$

$\sqrt[]{x^{5}}$

$\frac{1}{\sqrt[5]{x^{2}}}$

$5\sqrt{x}$

$2\sqrt[5]{x}$

Explanation

Fractional exponents are an alternate way to write a radical expression. The numerator of the fractional exponent becomes the exponent of the term inside the radical and the denominator of the fractional exponent determines the index of the radical.

So $x^{2/5}$ becomes $\sqrt[5]{x^{2}}$ .

Simplify: $3^{\frac{1}{8}}$

$\sqrt[8]{3}$

$\frac{1}{3^8}$

$\frac{1}{\sqrt[8]{3}}$

$\sqrt{3^8}$

$\frac{3}{8}$

Explanation

When dealing with fractional exponents, we rewrite as such: $x^{\frac{a}{b}}=\sqrt[b]{x^a}$ which is the index of the radical and is the exponent raising base .

$3^{\frac{1}{8}}=\sqrt[8]{3}$

Find the value of $64^{\frac{4}{3}}$

Explanation

When you have a number or value with a fractional exponent,

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

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

So then,

$64^{\frac{4}{3}}=\sqrt[3]{64^4}=\sqrt[3]{2^{24}}=2^8=256$

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.

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

Which of the following is equivalent to $b^{\frac{4}{5}}$ ?

$\sqrt[5]{b^4}$

$\sqrt[4]{b^5}$

$\sqrt[4]{b^{-5}}$

Explanation

Which of the following is equivalent to $b^{\frac{4}{5}}$ ?

When dealing with fractional exponents, keep the following in mind: The numerator is making the base bigger, so treat it like a regular exponent. The denominator is making the base smaller, so it must be the root you are taking.

This means that $b^{\frac{4}{5}}$ is equal to the fifth root of b to the fourth. Perhaps a bit confusing, but it means that we will keep , but put the whole thing under $\sqrt[5]{b}$ .