Simplify Rational Exponents

Simplify Expressions with

Rational exponents are another way of writing expressions with radicals. When we use rational exponents, we can apply the properties of exponents to simplify expressions.

The Power Property for Exponents says that \({({a}^{m})}^{n}={a}^{m\cdot n}\) when m and n are whole numbers. Let’s assume we are now not limited to whole numbers.

Suppose we want to find a number p such that \({({8}^{p})}^{3}=8.\) We will use the Power Property of Exponents to find the value of p.

\(\begin{array}{llllllll} & & & & & {({8}^{p})}^{3} & = & 8 \\ \text{Multiply the exponents on the left.} & & & & & {8}^{3p} & = & 8 \\ \text{Write the exponent 1 on the right.} & & & & & {8}^{3p} & = & {8}^{1} \\ \text{Since the bases are the same, the exponents must be equal.} & & & & & 3p & = & 1 \\ \text{Solve for}\ p. & & & & & p & = & \frac{1}{3}\end{array}\)

So \({({8}^{\frac{1}{3}})}^{3}=8.\) But we know also \({(\sqrt[3]{8})}^{3}=8.\) Then it must be that \({8}^{\frac{1}{3}}=\sqrt[3]{8}.\)

This same logic can be used for any positive integer exponent n to show that \({a}^{\frac{1}{n}}=\sqrt[n]{a}.\)

The denominator of the rational exponent is the index of the radical.

Example

Try it.

Write as a radical expression: ⓐ \({x}^{\frac{1}{2}}\) ⓑ \({y}^{\frac{1}{3}}\) ⓒ \({z}^{\frac{1}{4}}.\)

Solution

We want to write each expression in the form \(\sqrt[n]{a}.\)

ⓐ

	\(\ {x}^{\frac{1}{2}}\)
The denominator of the rational exponent is 2, so the index of the radical is 2. We do not show the index when it is 2.	\(\ \sqrt{x}\)

ⓑ

	\(\ {y}^{\frac{1}{3}}\)
The denominator of the exponent is 3, so the index is 3.	\(\ \sqrt[3]{y}\)

ⓒ

	\(\ {z}^{\frac{1}{4}}\)
The denominator of the exponent is 4, so the index is 4.	\(\ \sqrt[4]{z}\)

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Simplify Expressions with

We can look at \({a}^{\frac{m}{n}}\) in two ways. Remember the Power Property tells us to multiply the exponents and so \({({a}^{\frac{1}{n}})}^{m}\) and \({({a}^{m})}^{{}^{\frac{1}{n}}}\) both equal \({a}^{\frac{m}{n}}.\) If we write these expressions in radical form, we get

\[{a}^{\frac{m}{n}}={({a}^{\frac{1}{n}})}^{m}={(\sqrt[n]{a})}^{m}\ \text{and}\ {a}^{\frac{m}{n}}={({a}^{m})}^{{}^{\frac{1}{n}}}=\sqrt[n]{{a}^{m}}\]

This leads us to the following definition.

Which form do we use to simplify an expression? We usually take the root first—that way we keep the numbers in the radicand smaller, before raising it to the power indicated.

Example

Try it.

Write with a rational exponent: ⓐ \(\sqrt{{y}^{3}}\) ⓑ \({(\sqrt[3]{2x})}^{4}\) ⓒ \(\sqrt{{(\frac{3a}{4b})}^{3}}.\)

Solution

We want to use \({a}^{\frac{m}{n}}=\sqrt[n]{{a}^{m}}\) to write each radical in the form \({a}^{\frac{m}{n}}.\)

ⓐ

ⓑ

ⓒ

Remember that \({a}^{\text{-}n}=\frac{1}{{a}^{n}}.\) The negative sign in the exponent does not change the sign of the expression.

Example

Try it.

Simplify: ⓐ \(\text{-}{25}^{\frac{3}{2}}\) ⓑ \(\text{-}{25}^{-\frac{3}{2}}\) ⓒ \({(-25)}^{\frac{3}{2}}.\)

Solution

ⓐ

	\(\ \text{-}{25}^{\frac{3}{2}}\)
Rewrite in radical form.	\(\ \text{-}{(\sqrt{25})}^{3}\)
Simplify the radical.	\(\ \text{-}{(5)}^{3}\)
Simplify.	\(\ -125\)

ⓑ

	\(\ \text{-}{25}^{-\frac{3}{2}}\)
Rewrite using \({a}^{\text{-}n}=\frac{1}{{a}^{n}}.\)	\(\ \text{-}(\frac{1}{{25}^{\frac{3}{2}}})\)
Rewrite in radical form.	\(\ \text{-}(\frac{1}{{(\sqrt{25})}^{3}})\)
Simplify the radical.	\(\ \text{-}(\frac{1}{{(5)}^{3}})\)
Simplify.	\(\ -\frac{1}{125}\)

ⓒ

	\(\ {(-25)}^{\frac{3}{2}}\)
Rewrite in radical form.	\(\ {(\sqrt{-25})}^{3}\)
There is no real number whose square root \(\text{is}\ -25.\)	\(\ \text{Not a real number.}\)

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Use the Properties of Exponents to Simplify Expressions with Rational Exponents

The same properties of exponents that we have already used also apply to rational exponents. We will list the Properties of Exponenets here to have them for reference as we simplify expressions.

We will apply these properties in the next example.

Example

Try it.

Simplify: ⓐ \({x}^{\frac{1}{2}}\cdot {x}^{\frac{5}{6}}\) ⓑ \({({z}^{9})}^{\frac{2}{3}}\) ⓒ \(\frac{{x}^{\frac{1}{3}}}{{x}^{\frac{5}{3}}}.\)

Solution

ⓐ The Product Property tells us that when we multiply the same base, we add the exponents.

	\(\ {x}^{\frac{1}{2}}\cdot {x}^{\frac{5}{6}}\)
The bases are the same, so we add the exponents.	\(\ {x}^{\frac{1}{2}+\frac{5}{6}}\)
Add the fractions.	\(\ {x}^{\frac{8}{6}}\)
Simplify the exponent.	\(\ {x}^{\frac{4}{3}}\)

ⓑ The Power Property tells us that when we raise a power to a power, we multiply the exponents.

	\(\ {({z}^{9})}^{\frac{2}{3}}\)
To raise a power to a power, we multiply the exponents.	\(\ {z}^{9\cdot \frac{2}{3}}\)
Simplify.	\(\ {z}^{6}\)

ⓒ The Quotient Property tells us that when we divide with the same base, we subtract the exponents.

	\(\ \frac{{x}^{\frac{1}{3}}}{{x}^{\frac{5}{3}}}\)
	\(\ \frac{{x}^{\frac{1}{3}}}{{x}^{\frac{5}{3}}}\)
To divide with the same base, we subtract the exponents.	\(\ \frac{1}{{x}^{\frac{5}{3}-\frac{1}{3}}}\)
Simplify.	\(\ \frac{1}{{x}^{\frac{4}{3}}}\)

Sometimes we need to use more than one property. In the next example, we will use both the Product to a Power Property and then the Power Property.

We will use both the Product Property and the Quotient Property in the next example.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Key Concepts

• Rational Exponent \({a}^{\frac{1}{n}}\)
  • If \(\sqrt[n]{a}\) is a real number and \(n\ge 2,\) then \({a}^{\frac{1}{n}}=\sqrt[n]{a}.\)
• Rational Exponent \({a}^{\frac{m}{n}}\)
  • For any positive integers m and n,
    \({a}^{\frac{m}{n}}={(\sqrt[n]{a})}^{m}\) and \({a}^{\frac{m}{n}}=\sqrt[n]{{a}^{m}}\)
• Properties of Exponents
  • If a, b are real numbers and m, n are rational numbers, then
    • Product Property \({a}^{m}\cdot {a}^{n}={a}^{m+n}\)
    • Power Property \({({a}^{m})}^{n}={a}^{m\cdot n}\)
    • Product to a Power \({(ab)}^{m}={a}^{m}{b}^{m}\)
    • Quotient Property \(\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},\ a\ne 0\)
    • Zero Exponent Definition \({a}^{0}=1,\) \(a\ne 0\)
    • Quotient to a Power Property \({(\frac{a}{b})}^{m}=\frac{{a}^{m}}{{b}^{m}},\ b\ne 0\)
    • Negative Exponent Property \({a}^{\text{-}n}=\frac{1}{{a}^{n}},a\ne 0\)

Simplify Rational Exponents

Simplify expressions with \({a}^{\frac{1}{n}}\)

In the following exercises, write as a radical expression.

Try it.

ⓐ \({x}^{\frac{1}{2}}\) ⓑ \({y}^{\frac{1}{3}}\) ⓒ \({z}^{\frac{1}{4}}\)

Solution

ⓐ \(\sqrt{x}\) ⓑ \(\sqrt[3]{y}\) ⓒ \(\sqrt[4]{z}\)

Try it.

ⓐ \({r}^{\frac{1}{2}}\) ⓑ \({s}^{\frac{1}{3}}\) ⓒ \({t}^{\frac{1}{4}}\)

Try it.

ⓐ \({u}^{\frac{1}{5}}\) ⓑ \({v}^{\frac{1}{9}}\) ⓒ \({w}^{\frac{1}{20}}\)

Solution

ⓐ \(\sqrt[5]{u}\) ⓑ \(\sqrt[9]{v}\) ⓒ \(\sqrt[20]{w}\)

Try it.

ⓐ \({g}^{\frac{1}{7}}\) ⓑ \({h}^{\frac{1}{5}}\) ⓒ \({j}^{\frac{1}{25}}\)

In the following exercises, write with a rational exponent.

Try it.

ⓐ \(\sqrt[7]{x}\) ⓑ \(\sqrt[9]{y}\) ⓒ \(\sqrt[5]{f}\)

Solution

ⓐ \({x}^{\frac{1}{7}}\) ⓑ \({y}^{\frac{1}{9}}\) ⓒ \({f}^{\frac{1}{5}}\)

Try it.

ⓐ \(\sqrt[8]{r}\) ⓑ \(\sqrt[10]{s}\) ⓒ \(\sqrt[4]{t}\)

Try it.

ⓐ \(\sqrt[3]{7c}\) ⓑ \(\sqrt[7]{12d}\) ⓒ \(2\sqrt[4]{6b}\)

Solution

ⓐ \({(7c)}^{\frac{1}{3}}\) ⓑ \({(12d)}^{\frac{1}{7}}\)
ⓒ \(2{(6b)}^{\frac{1}{4}}\)

Try it.

ⓐ \(\sqrt[4]{5x}\) ⓑ \(\sqrt[8]{9y}\) ⓒ \(7\sqrt[5]{3z}\)

Try it.

ⓐ \(\sqrt{21p}\) ⓑ \(\sqrt[4]{8q}\) ⓒ \(4\sqrt[6]{36r}\)

Solution

ⓐ \({(21p)}^{\frac{1}{2}}\) ⓑ \({(8q)}^{\frac{1}{4}}\)
ⓒ \(4{(36r)}^{\frac{1}{6}}\)

Try it.

ⓐ \(\sqrt[3]{25a}\) ⓑ \(\sqrt{3b}\) ⓒ \(\sqrt[8]{40c}\)

In the following exercises, simplify.

Try it.

ⓐ \({81}^{\frac{1}{2}}\) ⓑ \({125}^{\frac{1}{3}}\) ⓒ \({64}^{\frac{1}{2}}\)

Solution

ⓐ 9 ⓑ 5 ⓒ 8

Try it.

ⓐ \({625}^{\frac{1}{4}}\) ⓑ \({243}^{\frac{1}{5}}\) ⓒ \({32}^{\frac{1}{5}}\)

Try it.

ⓐ \({16}^{\frac{1}{4}}\) ⓑ \({16}^{\frac{1}{2}}\) ⓒ \({625}^{\frac{1}{4}}\)

Solution

ⓐ 2 ⓑ 4 ⓒ 5

Try it.

ⓐ \({64}^{\frac{1}{3}}\) ⓑ \({32}^{\frac{1}{5}}\) ⓒ \({81}^{\frac{1}{4}}\)

Try it.

ⓐ \({(-216)}^{\frac{1}{3}}\) ⓑ \(\text{-}{216}^{\frac{1}{3}}\) ⓒ \({(216)}^{-\frac{1}{3}}\)

Solution

ⓐ \(-6\) ⓑ \(-6\) ⓒ \(\frac{1}{6}\)

Try it.

ⓐ \({(-1000)}^{\frac{1}{3}}\) ⓑ \(\text{-}{1000}^{\frac{1}{3}}\) ⓒ \({(1000)}^{-\frac{1}{3}}\)

Try it.

ⓐ \({(-81)}^{\frac{1}{4}}\) ⓑ \(\text{-}{81}^{\frac{1}{4}}\) ⓒ \({(81)}^{-\frac{1}{4}}\)

Solution

ⓐ not real ⓑ \(-3\) ⓒ \(\frac{1}{3}\)

Try it.

ⓐ \({(-49)}^{\frac{1}{2}}\) ⓑ \(\text{-}{49}^{\frac{1}{2}}\) ⓒ \({(49)}^{-\frac{1}{2}}\)

Try it.

ⓐ \({(-36)}^{\frac{1}{2}}\) ⓑ \(\text{-}{36}^{\frac{1}{2}}\) ⓒ \({(36)}^{-\frac{1}{2}}\)

Solution

ⓐ not real ⓑ \(-6\) ⓒ \(\frac{1}{6}\)

Try it.

ⓐ \({(-16)}^{\frac{1}{4}}\) ⓑ \(\text{-}{16}^{\frac{1}{4}}\) ⓒ \({16}^{-\frac{1}{4}}\)

Try it.

ⓐ \({(-100)}^{\frac{1}{2}}\) ⓑ \(\text{-}{100}^{\frac{1}{2}}\) ⓒ \({(100)}^{-\frac{1}{2}}\)

Solution

ⓐ not real ⓑ \(-10\) ⓒ \(\frac{1}{10}\)

Try it.

ⓐ \({(-32)}^{\frac{1}{5}}\) ⓑ \({(243)}^{-\frac{1}{5}}\) ⓒ \(\text{-}{125}^{\frac{1}{3}}\)

Simplify Expressions with \({a}^{\frac{m}{n}}\)

In the following exercises, write with a rational exponent.

Try it.

ⓐ \(\sqrt{{m}^{5}}\) ⓑ \({(\sqrt[3]{3y})}^{7}\) ⓒ \(\sqrt[5]{{(\frac{4x}{5y})}^{3}}\)

Solution

ⓐ \({m}^{\frac{5}{2}}\) ⓑ \({(3y)}^{\frac{7}{3}}\) ⓒ \({(\frac{4x}{5y})}^{\frac{3}{5}}\)

Try it.

ⓐ \(\sqrt[4]{{r}^{7}}\) ⓑ \({(\sqrt[5]{2pq})}^{3}\) ⓒ \(\sqrt[4]{{(\frac{12m}{7n})}^{3}}\)

Try it.

ⓐ \(\sqrt[5]{{u}^{2}}\) ⓑ \({(\sqrt[3]{6x})}^{5}\) ⓒ \(\sqrt[4]{{(\frac{18a}{5b})}^{7}}\)

Solution

ⓐ \({u}^{\frac{2}{5}}\) ⓑ \({(6x)}^{\frac{5}{3}}\) ⓒ \({(\frac{18a}{5b})}^{\frac{7}{4}}\)

Try it.

ⓐ \(\sqrt[3]{a}\) ⓑ \({(\sqrt[4]{21v})}^{3}\) ⓒ \(\sqrt[4]{{(\frac{2xy}{5z})}^{2}}\)

In the following exercises, simplify.

Try it.

ⓐ \({64}^{\frac{5}{2}}\) ⓑ \({81}^{\frac{-3}{2}}\) ⓒ \({(-27)}^{\frac{2}{3}}\)

Solution

ⓐ 32,768 ⓑ \(\frac{1}{729}\) ⓒ 9

Try it.

ⓐ \({25}^{\frac{3}{2}}\) ⓑ \({9}^{-\frac{3}{2}}\) ⓒ \({(-64)}^{\frac{2}{3}}\)

Try it.

ⓐ \({32}^{\frac{2}{5}}\) ⓑ \({27}^{-\frac{2}{3}}\) ⓒ \({(-25)}^{\frac{1}{2}}\)

Solution

ⓐ 4 ⓑ \(\frac{1}{9}\) ⓒ not real

Try it.

ⓐ \({100}^{\frac{3}{2}}\) ⓑ \({49}^{-\frac{5}{2}}\) ⓒ \({(-100)}^{\frac{3}{2}}\)

Try it.

ⓐ \(\text{-}{9}^{\frac{3}{2}}\) ⓑ \(\text{-}{9}^{-\frac{3}{2}}\) ⓒ \({(-9)}^{\frac{3}{2}}\)

Solution

ⓐ \(-27\) ⓑ \(-\frac{1}{27}\) ⓒ not real

Try it.

ⓐ \(\text{-}{64}^{\frac{3}{2}}\) ⓑ \(\text{-}{64}^{-\frac{3}{2}}\) ⓒ \({(-64)}^{\frac{3}{2}}\)

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify. Assume all variables are positive.

Try it.

ⓐ \({c}^{\frac{1}{4}}\cdot {c}^{\frac{5}{8}}\) ⓑ \({({p}^{12})}^{\frac{3}{4}}\) ⓒ \(\frac{{r}^{\frac{4}{5}}}{{r}^{\frac{9}{5}}}\)

Solution

ⓐ \({c}^{\frac{7}{8}}\) ⓑ \({p}^{9}\) ⓒ \(\frac{1}{r}\)

Try it.

ⓐ \({6}^{\frac{5}{2}}\cdot {6}^{\frac{1}{2}}\) ⓑ \({({b}^{15})}^{\frac{3}{5}}\) ⓒ \(\frac{{w}^{\frac{2}{7}}}{{w}^{\frac{9}{7}}}\)

Try it.

ⓐ \({y}^{\frac{1}{2}}\cdot {y}^{\frac{3}{4}}\) ⓑ \({({x}^{12})}^{\frac{2}{3}}\) ⓒ \(\frac{{m}^{\frac{5}{8}}}{{m}^{\frac{13}{8}}}\)

Solution

ⓐ \({y}^{\frac{5}{4}}\) ⓑ \({x}^{8}\) ⓒ \(\frac{1}{m}\)

Try it.

ⓐ \({q}^{\frac{2}{3}}\cdot {q}^{\frac{5}{6}}\) ⓑ \({({h}^{6})}^{\frac{4}{3}}\) ⓒ \(\frac{{n}^{\frac{3}{5}}}{{n}^{\frac{8}{5}}}\)

Try it.

ⓐ \({(27{q}^{\frac{3}{2}})}^{\frac{4}{3}}\) ⓑ \({({a}^{\frac{1}{3}}{b}^{\frac{2}{3}})}^{\frac{3}{2}}\)

Solution

ⓐ \(81{q}^{2}\) ⓑ \({a}^{\frac{1}{2}}b\)

Try it.

ⓐ \({(64{s}^{\frac{3}{7}})}^{\frac{1}{6}}\) ⓑ \({({m}^{\frac{4}{3}}{n}^{\frac{1}{2}})}^{\frac{3}{4}}\)

Try it.

ⓐ \({(16\ {u}^{\frac{1}{3}})}^{\frac{3}{4}}\) ⓑ \({(4\ {p}^{\frac{1}{3}}{q}^{\frac{1}{2}})}^{\frac{3}{2}}\)

Solution

ⓐ \(8{u}^{\frac{1}{4}}\) ⓑ \(8{p}^{\frac{1}{2}}{q}^{\frac{3}{4}}\)

Try it.

ⓐ \({(625\ {n}^{\frac{8}{3}})}^{\frac{3}{4}}\) ⓑ \({(9\ {x}^{\frac{2}{5}}{y}^{\frac{3}{5}})}^{\frac{5}{2}}\)

Try it.

ⓐ \(\frac{{r}^{\frac{5}{2}}\cdot {r}^{-\frac{1}{2}}}{{r}^{-\frac{3}{2}}}\) ⓑ \({(\frac{36\ {s}^{\frac{1}{5}}{t}^{-\frac{3}{2}}}{{s}^{-\frac{9}{5}}{t}^{\frac{1}{2}}})}^{\frac{1}{2}}\)

Solution

ⓐ \({r}^{\frac{7}{2}}\) ⓑ \(\frac{6s}{t}\)

Try it.

ⓐ \(\frac{{a}^{\frac{3}{4}}\cdot {a}^{-\frac{1}{4}}}{{a}^{-\frac{10}{4}}}\) ⓑ \({(\frac{27\ {b}^{\frac{2}{3}}{c}^{-\frac{5}{2}}}{{b}^{-\frac{7}{3}}{c}^{\frac{1}{2}}})}^{\frac{1}{3}}\)

Try it.

ⓐ \(\frac{{c}^{\frac{5}{3}}\cdot {c}^{-\frac{1}{3}}}{{c}^{-\frac{2}{3}}}\) ⓑ \({(\frac{8\ {x}^{\frac{5}{3}}\ {y}^{-\frac{1}{2}}}{27\ {x}^{-\frac{4}{3}}\ {y}^{\frac{5}{2}}})}^{\frac{1}{3}}\)

Solution

ⓐ \({c}^{2}\) ⓑ \(\frac{2x}{3y}\)

Try it.

ⓐ \(\frac{{m}^{\frac{7}{4}}\cdot {m}^{-\frac{5}{4}}}{{m}^{-\frac{2}{4}}}\) ⓑ \({(\frac{16\ {m}^{\frac{1}{5}}\ {n}^{\frac{3}{2}}}{81\ {m}^{\frac{9}{5}}\ {n}^{-\frac{1}{2}}})}^{\frac{1}{4}}\)

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

1. Add: \(\frac{7}{15}+\frac{5}{12}.\)
   If you missed this problem, review .

   Revelar a resposta

   \(\frac{53}{60}\)

2. Simplify: \({(4{x}^{2}{y}^{5})}^{3}.\)
   If you missed this problem, review .

   Revelar a resposta

   \(64{x}^{6}{y}^{15}\)

3. Simplify: \({5}^{-3}.\)
   If you missed this problem, review .

   Revelar a resposta

   \(\frac{1}{125}\)

4. Write as a radical expression: ⓐ \({x}^{\frac{1}{2}}\) ⓑ \({y}^{\frac{1}{3}}\) ⓒ \({z}^{\frac{1}{4}}.\)

   Revelar a resposta

   We want to write each expression in the form \(\sqrt[n]{a}.\)

   ⓐ
   

   	\(\ {x}^{\frac{1}{2}}\)
   The denominator of the rational exponent is 2, so the index of the radical is 2. We do not show the index when it is 2.	\(\ \sqrt{x}\)

   ⓑ
   

   	\(\ {y}^{\frac{1}{3}}\)
   The denominator of the exponent is 3, so the index is 3.	\(\ \sqrt[3]{y}\)

   ⓒ
   

   	\(\ {z}^{\frac{1}{4}}\)
   The denominator of the exponent is 4, so the index is 4.	\(\ \sqrt[4]{z}\)

5. Write as a radical expression: ⓐ \({t}^{\frac{1}{2}}\) ⓑ \({m}^{\frac{1}{3}}\) ⓒ \({r}^{\frac{1}{4}}.\)

   Revelar a resposta

   ⓐ \(\sqrt{t}\) ⓑ \(\sqrt[3]{m}\) ⓒ \(\sqrt[4]{r}\)

6. Write as a radial expression: ⓐ \({b}^{\frac{1}{6}}\) ⓑ \({z}^{\frac{1}{5}}\) ⓒ \({p}^{\frac{1}{4}}.\)

   Revelar a resposta

   ⓐ \(\sqrt[6]{b}\) ⓑ \(\sqrt[5]{z}\) ⓒ \(\sqrt[4]{p}\)

7. Write with a rational exponent: ⓐ \(\sqrt{5y}\) ⓑ \(\sqrt[3]{4x}\) ⓒ \(3\sqrt[4]{5z}.\)

   Revelar a resposta

   We want to write each radical in the form \({a}^{\frac{1}{n}}.\)

   ⓐ
   

   	\(\ \sqrt{5y}\)
   No index is shown, so it is 2. The denominator of the exponent will be 2.	\(\ {(5y)}^{\frac{1}{2}}\)
   Put parentheses around the entire expression \(5y.\)

   ⓑ
   

   	\(\ \sqrt[3]{4x}\)
   The index is 3, so the denominator of the exponent is 3. Include parentheses \((4x).\)	\(\ {(4x)}^{\frac{1}{3}}\)

   ⓒ
   

   	\(\ 3\ \sqrt[4]{5z}\)
   The index is 4, so the denominator of the exponent is 4. Put parentheses only around the \(5z\) since 3 is not under the radical sign.	\(\ 3{(5z)}^{\frac{1}{4}}\)

8. Write with a rational exponent: ⓐ \(\sqrt{10m}\) ⓑ \(\sqrt[5]{3n}\) ⓒ \(3\sqrt[4]{6y}.\)

   Revelar a resposta

   ⓐ \({(10m)}^{\frac{1}{2}}\) ⓑ \({(3n)}^{\frac{1}{5}}\)
   ⓒ \(3{(6y)}^{\frac{1}{4}}\)

9. Write with a rational exponent: ⓐ \(\sqrt[7]{3k}\) ⓑ \(\sqrt[4]{5j}\) ⓒ \(8\sqrt[3]{2a}.\)

   Revelar a resposta

   ⓐ \({(3k)}^{\frac{1}{7}}\) ⓑ \({(5j)}^{\frac{1}{4}}\)
   ⓒ \(8{(2a)}^{\frac{1}{3}}\)

10. Simplify: ⓐ \({25}^{\frac{1}{2}}\) ⓑ \({64}^{\frac{1}{3}}\) ⓒ \({256}^{\frac{1}{4}}.\)

    Revelar a resposta

    ⓐ
    

    	\(\ {25}^{\frac{1}{2}}\)
    Rewrite as a square root.	\(\ \sqrt{25}\)
    Simplify.	\(\ 5\)

    ⓑ
    

    	\(\ {64}^{\frac{1}{3}}\)
    Rewrite as a cube root.	\(\ \sqrt[3]{64}\)
    Recognize 64 is a perfect cube.	\(\ \sqrt[3]{{4}^{3}}\)
    Simplify.	\(\ 4\)

    ⓒ
    

    	\(\ {256}^{\frac{1}{4}}\)
    Rewrite as a fourth root.	\(\ \sqrt[4]{256}\)
    Recognize 256 is a perfect fourth power.	\(\ \sqrt[4]{{4}^{4}}\)
    Simplify.	\(\ 4\)

11. Simplify: ⓐ \({36}^{\frac{1}{2}}\) ⓑ \({8}^{\frac{1}{3}}\) ⓒ \({16}^{\frac{1}{4}}.\)

    Revelar a resposta

    ⓐ 6 ⓑ 2 ⓒ 2

12. Simplify: ⓐ \({100}^{\frac{1}{2}}\) ⓑ \({27}^{\frac{1}{3}}\) ⓒ \({81}^{\frac{1}{4}}.\)

    Revelar a resposta

    ⓐ 10 ⓑ 3 ⓒ 3

13. Simplify: ⓐ \({(-16)}^{\frac{1}{4}}\) ⓑ \(\text{-}{16}^{\frac{1}{4}}\) ⓒ \({(16)}^{-\frac{1}{4}}.\)

    Revelar a resposta

    ⓐ
    

    	\(\ {(-16)}^{\frac{1}{4}}\)
    Rewrite as a fourth root.	\(\ \sqrt[4]{-16}\)
    	\(\ \sqrt[4]{{(-2)}^{4}}\)
    Simplify.	\(\ \text{No real solution.}\)

    ⓑ
    

    	\(\ \text{-}{16}^{\frac{1}{4}}\)
    The exponent only applies to the 16. Rewrite as a fouth root.	\(\ \text{-}\sqrt[4]{16}\)
    Rewrite 16 as \({2}^{4}.\)	\(\ \text{-}\sqrt[4]{{2}^{4}}\)
    Simplify.	\(\ -2\)

    ⓒ
    

    	\(\ {(16)}^{-\frac{1}{4}}\)
    Rewrite using the property \({a}^{\text{-}n}=\frac{1}{{a}^{n}}.\)	\(\ \frac{1}{{(16)}^{\frac{1}{4}}}\)
    Rewrite as a fourth root.	\(\ \frac{1}{\sqrt[4]{16}}\)
    Rewrite 16 as \({2}^{4}.\)	\(\ \frac{1}{\sqrt[4]{{2}^{4}}}\)
    Simplify.	\(\ \frac{1}{2}\)

14. Simplify: ⓐ \({(-64)}^{-\frac{1}{2}}\) ⓑ \(\text{-}{64}^{\frac{1}{2}}\) ⓒ \({(64)}^{-\frac{1}{2}}.\)

    Revelar a resposta

    ⓐ No real solution ⓑ \(-8\)
    ⓒ \(\frac{1}{8}\)

15. Simplify: ⓐ \({(-256)}^{\frac{1}{4}}\) ⓑ \(\text{-}{256}^{\frac{1}{4}}\) ⓒ \({(256)}^{-\frac{1}{4}}.\)

    Revelar a resposta

    ⓐ No real solution ⓑ \(-4\)
    ⓒ \(\frac{1}{4}\)

16. Write with a rational exponent: ⓐ \(\sqrt{{y}^{3}}\) ⓑ \({(\sqrt[3]{2x})}^{4}\) ⓒ \(\sqrt{{(\frac{3a}{4b})}^{3}}.\)

    Revelar a resposta

    We want to use \({a}^{\frac{m}{n}}=\sqrt[n]{{a}^{m}}\) to write each radical in the form \({a}^{\frac{m}{n}}.\)

    ⓐ
    

    
    
    ⓑ
    

    
    
    ⓒ
    

17. Write with a rational exponent: ⓐ \(\sqrt{{x}^{5}}\) ⓑ \({(\sqrt[4]{3y})}^{3}\) ⓒ \(\sqrt{{(\frac{2m}{3n})}^{5}}.\)

    Revelar a resposta

    ⓐ \({x}^{\frac{5}{2}}\) ⓑ \({(3y)}^{\frac{3}{4}}\) ⓒ \({(\frac{2m}{3n})}^{\frac{5}{2}}\)

18. Write with a rational exponent: ⓐ \(\sqrt[5]{{a}^{2}}\) ⓑ \({(\sqrt[3]{5ab})}^{5}\) ⓒ \(\sqrt{{(\frac{7xy}{z})}^{3}}.\)

    Revelar a resposta

    ⓐ \({a}^{\frac{2}{5}}\) ⓑ \({(5ab)}^{\frac{5}{3}}\)
    ⓒ \({(\frac{7xy}{z})}^{\frac{3}{2}}\)

19. Simplify: ⓐ \({125}^{\frac{2}{3}}\) ⓑ \({16}^{-\frac{3}{2}}\) ⓒ \({32}^{-\frac{2}{5}}.\)

    Revelar a resposta

    We will rewrite the expression as a radical first using the defintion, \({a}^{\frac{m}{n}}={(\sqrt[n]{a})}^{m}.\) This form lets us take the root first and so we keep the numbers in the radicand smaller than if we used the other form.

    ⓐ
    

    	\(\ {125}^{\frac{2}{3}}\)
    The power of the radical is the numerator of the exponent, 2. The index of the radical is the denominator of the exponent, 3.	\(\ {(\sqrt[3]{125})}^{2}\)
    Simplify.	\(\ {(5)}^{2}\)
    	\(\ 25\)

    ⓑ We will rewrite each expression first using \({a}^{\text{-}n}=\frac{1}{{a}^{n}}\) and then change to radical form.
    

    	\(\ {16}^{-\frac{3}{2}}\)
    Rewrite using \({a}^{\text{-}n}=\frac{1}{{a}^{n}}\)	\(\ \frac{1}{{16}^{\frac{3}{2}}}\)
    Change to radical form. The power of the radical is the numerator of the exponent, 3. The index is the denominator of the exponent, 2.	\(\ \frac{1}{{(\sqrt{16})}^{3}}\)
    Simplify.	\(\ \frac{1}{{4}^{3}}\)
    	\(\ \frac{1}{64}\)

    ⓒ
    

    	\(\ {32}^{-\frac{2}{5}}\)
    Rewrite using \({a}^{\text{-}n}=\frac{1}{{a}^{n}}.\)	\(\ \frac{1}{{32}^{\frac{2}{5}}}\)
    Change to radical form.	\(\ \frac{1}{{(\sqrt[5]{32})}^{2}}\)
    Rewrite the radicand as a power.	\(\ \frac{1}{{(\sqrt[5]{{2}^{5}})}^{2}}\)
    Simplify.	\(\ \frac{1}{{2}^{2}}\)
    	\(\ \frac{1}{4}\)

20. Simplify: ⓐ \({27}^{\frac{2}{3}}\) ⓑ \({81}^{-\frac{3}{2}}\) ⓒ \({16}^{-\frac{3}{4}}.\)

    Revelar a resposta

    ⓐ 9 ⓑ \(\frac{1}{729}\) ⓒ \(\frac{1}{8}\)

21. Simplify: ⓐ \({4}^{\frac{3}{2}}\) ⓑ \({27}^{-\frac{2}{3}}\) ⓒ \({625}^{-\frac{3}{4}}.\)

    Revelar a resposta

    ⓐ 8 ⓑ \(\frac{1}{9}\) ⓒ \(\frac{1}{125}\)

22. Simplify: ⓐ \(\text{-}{25}^{\frac{3}{2}}\) ⓑ \(\text{-}{25}^{-\frac{3}{2}}\) ⓒ \({(-25)}^{\frac{3}{2}}.\)

    Revelar a resposta

    ⓐ
    

    	\(\ \text{-}{25}^{\frac{3}{2}}\)
    Rewrite in radical form.	\(\ \text{-}{(\sqrt{25})}^{3}\)
    Simplify the radical.	\(\ \text{-}{(5)}^{3}\)
    Simplify.	\(\ -125\)

    ⓑ
    

    	\(\ \text{-}{25}^{-\frac{3}{2}}\)
    Rewrite using \({a}^{\text{-}n}=\frac{1}{{a}^{n}}.\)	\(\ \text{-}(\frac{1}{{25}^{\frac{3}{2}}})\)
    Rewrite in radical form.	\(\ \text{-}(\frac{1}{{(\sqrt{25})}^{3}})\)
    Simplify the radical.	\(\ \text{-}(\frac{1}{{(5)}^{3}})\)
    Simplify.	\(\ -\frac{1}{125}\)

    ⓒ
    

    	\(\ {(-25)}^{\frac{3}{2}}\)
    Rewrite in radical form.	\(\ {(\sqrt{-25})}^{3}\)
    There is no real number whose square root \(\text{is}\ -25.\)	\(\ \text{Not a real number.}\)

23. Simplify: ⓐ \({-16}^{\frac{3}{2}}\) ⓑ \({-16}^{-\frac{3}{2}}\) ⓒ \({(-16)}^{-\frac{3}{2}}.\)

    Revelar a resposta

    ⓐ \(-64\) ⓑ \(-\frac{1}{64}\) ⓒ not a real number

24. Simplify: ⓐ \({-81}^{\frac{3}{2}}\) ⓑ \({-81}^{-\frac{3}{2}}\) ⓒ \({(-81)}^{-\frac{3}{2}}.\)

    Revelar a resposta

    ⓐ \(-729\) ⓑ \(-\frac{1}{729}\) ⓒ not a real number

25. Simplify: ⓐ \({x}^{\frac{1}{2}}\cdot {x}^{\frac{5}{6}}\) ⓑ \({({z}^{9})}^{\frac{2}{3}}\) ⓒ \(\frac{{x}^{\frac{1}{3}}}{{x}^{\frac{5}{3}}}.\)

    Revelar a resposta

    ⓐ The Product Property tells us that when we multiply the same base, we add the exponents.

    	\(\ {x}^{\frac{1}{2}}\cdot {x}^{\frac{5}{6}}\)
    The bases are the same, so we add the exponents.	\(\ {x}^{\frac{1}{2}+\frac{5}{6}}\)
    Add the fractions.	\(\ {x}^{\frac{8}{6}}\)
    Simplify the exponent.	\(\ {x}^{\frac{4}{3}}\)

    ⓑ The Power Property tells us that when we raise a power to a power, we multiply the exponents.

    	\(\ {({z}^{9})}^{\frac{2}{3}}\)
    To raise a power to a power, we multiply the exponents.	\(\ {z}^{9\cdot \frac{2}{3}}\)
    Simplify.	\(\ {z}^{6}\)

    ⓒ The Quotient Property tells us that when we divide with the same base, we subtract the exponents.

    	\(\ \frac{{x}^{\frac{1}{3}}}{{x}^{\frac{5}{3}}}\)
    	\(\ \frac{{x}^{\frac{1}{3}}}{{x}^{\frac{5}{3}}}\)
    To divide with the same base, we subtract the exponents.	\(\ \frac{1}{{x}^{\frac{5}{3}-\frac{1}{3}}}\)
    Simplify.	\(\ \frac{1}{{x}^{\frac{4}{3}}}\)

26. Simplify: ⓐ \({x}^{\frac{1}{6}}\cdot {x}^{\frac{4}{3}}\) ⓑ \({({x}^{6})}^{\frac{4}{3}}\) ⓒ \(\frac{{x}^{\frac{2}{3}}}{{x}^{\frac{5}{3}}}.\)

    Revelar a resposta

    ⓐ \({x}^{\frac{3}{2}}\) ⓑ \({x}^{8}\) ⓒ \(\frac{1}{x}\)

27. Simplify: ⓐ \({y}^{\frac{3}{4}}\cdot {y}^{\frac{5}{8}}\) ⓑ \({({m}^{9})}^{\frac{2}{9}}\) ⓒ \(\frac{{d}^{\frac{1}{5}}}{{d}^{\frac{6}{5}}}.\)

    Revelar a resposta

    ⓐ \({y}^{\frac{11}{8}}\) ⓑ \({m}^{2}\) ⓒ \(\frac{1}{d}\)

28. Simplify: ⓐ \({(27{u}^{\frac{1}{2}})}^{\frac{2}{3}}\) ⓑ \({({m}^{\frac{2}{3}}{n}^{\frac{1}{2}})}^{\frac{3}{2}}.\)

    Revelar a resposta

    ⓐ
    

    	\(\ {(27{u}^{\frac{1}{2}})}^{\frac{2}{3}}\)
    First we use the Product to a Power Property.	\(\ {(27)}^{\frac{2}{3}}{({u}^{\frac{1}{2}})}^{\frac{2}{3}}\)
    Rewrite 27 as a power of 3.	\(\ {({3}^{3})}^{\frac{2}{3}}{({u}^{\frac{1}{2}})}^{\frac{2}{3}}\)
    To raise a power to a power, we multiply the exponents.	\(\ ({3}^{2})({u}^{\frac{1}{3}})\)
    Simplify.	\(\ 9{u}^{\frac{1}{3}}\)

    ⓑ
    

    	\(\ {({m}^{\frac{2}{3}}{n}^{\frac{1}{2}})}^{\frac{3}{2}}\)
    First we use the Product to a Power Property.	\(\ {({m}^{\frac{2}{3}})}^{\frac{3}{2}}{({n}^{\frac{1}{2}})}^{\frac{3}{2}}\)
    To raise a power to a power, we multiply the exponents.	\(\ m{n}^{\frac{3}{4}}\)

29. Simplify: ⓐ \({(32{x}^{\frac{1}{3}})}^{\frac{3}{5}}\) ⓑ \({({x}^{\frac{3}{4}}{y}^{\frac{1}{2}})}^{\frac{2}{3}}.\)

    Revelar a resposta

    ⓐ \(8{x}^{\frac{1}{5}}\) ⓑ \({x}^{\frac{1}{2}}{y}^{\frac{1}{3}}\)

30. Simplify: ⓐ \({(81{n}^{\frac{2}{5}})}^{\frac{3}{2}}\) ⓑ \({({a}^{\frac{3}{2}}{b}^{\frac{1}{2}})}^{\frac{4}{3}}.\)

    Revelar a resposta

    ⓐ \(729{n}^{\frac{3}{5}}\) ⓑ \({a}^{2}{b}^{\frac{2}{3}}\)

31. Simplify: ⓐ \(\frac{{x}^{\frac{3}{4}}\cdot {x}^{-\frac{1}{4}}}{{x}^{-\frac{6}{4}}}\) ⓑ \({(\frac{16\ {x}^{\frac{4}{3}}{y}^{-\frac{5}{6}}}{{x}^{-\frac{2}{3}}{y}^{\frac{1}{6}}})}^{\frac{1}{2}}.\)

    Revelar a resposta

    ⓐ
    

    	\(\ \frac{{x}^{\frac{3}{4}}\cdot {x}^{-\frac{1}{4}}}{{x}^{-\frac{6}{4}}}\)
    Use the Product Property in the numerator, add the exponents.	\(\ \frac{{x}^{\frac{2}{4}}}{{x}^{-\frac{6}{4}}}\)
    Use the Quotient Property, subtract the exponents.	\(\ {x}^{\frac{8}{4}}\)
    Simplify.	\(\ {x}^{2}\)

    ⓑ Follow the order of operations to simplify inside the parenthese first.

    	\(\ {(\frac{16\ {x}^{\frac{4}{3}}{y}^{-\frac{5}{6}}}{{x}^{-\frac{2}{3}}{y}^{\frac{1}{6}}})}^{\frac{1}{2}}\)
    Use the Quotient Property, subtract the exponents.	\(\ {(\frac{16{x}^{\frac{6}{3}}}{{y}^{\frac{6}{6}}})}^{\frac{1}{2}}\)
    Simplify.	\(\ {(\frac{16{x}^{2}}{y})}^{\frac{1}{2}}\)
    Use the Product to a Power Property, multiply the exponents.	\(\ \frac{4x}{{y}^{\frac{1}{2}}}\)

32. Simplify: ⓐ \(\frac{{m}^{\frac{2}{3}}\cdot {m}^{-\frac{1}{3}}}{{m}^{-\frac{5}{3}}}\) ⓑ \({(\frac{25{m}^{\frac{1}{6}}{n}^{\frac{11}{6}}}{{m}^{\frac{2}{3}}{n}^{-\frac{1}{6}}})}^{\frac{1}{2}}.\)

    Revelar a resposta

    ⓐ \({m}^{2}\) ⓑ \(\frac{5n}{{m}^{\frac{1}{4}}}\)

33. Simplify: ⓐ \(\frac{{u}^{\frac{4}{5}}\cdot {u}^{-\frac{2}{5}}}{{u}^{-\frac{13}{5}}}\) ⓑ \({(\frac{27{x}^{\frac{4}{5}}{y}^{\frac{1}{6}}}{{x}^{\frac{1}{5}}{y}^{-\frac{5}{6}}})}^{\frac{1}{3}}.\)

    Revelar a resposta

    ⓐ \({u}^{3}\) ⓑ \(3{x}^{\frac{1}{5}}{y}^{\frac{1}{3}}\)

34. ⓐ \({x}^{\frac{1}{2}}\) ⓑ \({y}^{\frac{1}{3}}\) ⓒ \({z}^{\frac{1}{4}}\)

    Revelar a resposta

    ⓐ \(\sqrt{x}\) ⓑ \(\sqrt[3]{y}\) ⓒ \(\sqrt[4]{z}\)

35. ⓐ \({r}^{\frac{1}{2}}\) ⓑ \({s}^{\frac{1}{3}}\) ⓒ \({t}^{\frac{1}{4}}\)

36. ⓐ \({u}^{\frac{1}{5}}\) ⓑ \({v}^{\frac{1}{9}}\) ⓒ \({w}^{\frac{1}{20}}\)

    Revelar a resposta

    ⓐ \(\sqrt[5]{u}\) ⓑ \(\sqrt[9]{v}\) ⓒ \(\sqrt[20]{w}\)

37. ⓐ \({g}^{\frac{1}{7}}\) ⓑ \({h}^{\frac{1}{5}}\) ⓒ \({j}^{\frac{1}{25}}\)

38. ⓐ \(\sqrt[7]{x}\) ⓑ \(\sqrt[9]{y}\) ⓒ \(\sqrt[5]{f}\)

    Revelar a resposta

    ⓐ \({x}^{\frac{1}{7}}\) ⓑ \({y}^{\frac{1}{9}}\) ⓒ \({f}^{\frac{1}{5}}\)

39. ⓐ \(\sqrt[8]{r}\) ⓑ \(\sqrt[10]{s}\) ⓒ \(\sqrt[4]{t}\)

40. ⓐ \(\sqrt[3]{7c}\) ⓑ \(\sqrt[7]{12d}\) ⓒ \(2\sqrt[4]{6b}\)

    Revelar a resposta

    ⓐ \({(7c)}^{\frac{1}{3}}\) ⓑ \({(12d)}^{\frac{1}{7}}\)
    ⓒ \(2{(6b)}^{\frac{1}{4}}\)

Symbols used here

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\neq

not equal
The two sides are different.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

i

imaginary unit
i² = −1.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Simplify Rational Exponents

1. Simplify expressions with
2. Simplify expressions with
3. Use the properties of exponents to simplify expressions with rational exponents
4. If
5. For any positive integers
6. If

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

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Parts of this page are adapted from OpenStax Intermediate Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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