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Rational Exponents

Simplify expressions with

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.

	\(\ \begin{array}{lll} \\ {({8}^{p})}^{3} & = & 8\end{array}\)
Multiply the exponents on the left.	\(\ \begin{array}{lll} \\ {8}^{3p} & = & 8\end{array}\)
Write the exponent 1 on the right.	\(\ \begin{array}{lll} \\ {8}^{3p} & = & {8}^{1}\end{array}\)
The exponents must be equal.	\(\ \begin{array}{lll} \\ 3p & = & 1\end{array}\)
Solve for \(p.\)	\(\ \begin{array}{lll} \\ p & = & \frac{1}{3}\end{array}\)
	\(\begin{array}{lll} \\ \\ \\ \text{So}\ {({8}^{\frac{1}{3}})}^{3} & = & 8.\end{array}\)
But we know also \({(\sqrt[3]{8})}^{3}=8\). Then it must be that \({8}^{\frac{1}{3}}=\sqrt[3]{8}\).

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 \({a}^{\frac{1}{n}}=\sqrt[n]{a}\).

There will be times when working with expressions will be easier if you use rational exponents and times when it will be easier if you use radicals. In the first few examples, you’ll practice converting expressions between these two notations.

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 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 Elementary Algebra 2e.

Simplify Expressions with

Let’s work with the Power Property for Exponents some more.

Suppose we raise \({a}^{\frac{1}{n}}\) to the power m.

	\({({a}^{\frac{1}{n}})}^{m}\)
Multiply the exponents.	\({a}^{\frac{1}{n}}{}^{\cdot m}\)
Simplify.	\({a}^{\frac{m}{n}}\)
So \({a}^{\frac{m}{n}}={(\sqrt[n]{a})}^{m}.\)

Now suppose we take \({a}^{m}\) to the \(\frac{1}{n}\) power.

	\({({a}^{m})}^{\frac{1}{n}}\)
Multiply the exponents.	\({a}^{m\cdot \frac{1}{n}}\)
Simplify.	\({a}^{\frac{m}{n}}\)
So \({a}^{\frac{m}{n}}=\sqrt[n]{{a}^{m}}\) also.

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.

Example

Try it.

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

Solution

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

Example

Try it.

Simplify: ⓐ \({9}^{\frac{3}{2}}\) ⓑ \({125}^{\frac{2}{3}}\) ⓒ \({81}^{\frac{3}{4}}\).

Solution

We will rewrite each expression as a radical first using the property, \({a}^{\frac{m}{n}}={(\sqrt[n]{a})}^{m}\). 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.

ⓐ

	\({9}^{\frac{3}{2}}\)
The power of the radical is the numerator of the exponent, 3. Since the denominator of the exponent is 2, this is a square root.	\({(\sqrt{9})}^{3}\)
Simplify.	\({(3)}^{3}\)
	\(27\)

ⓑ

	\({125}^{\frac{2}{3}}\)
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.	\({(5)}^{2}\)
	\(25\)

ⓒ

	\({81}^{\frac{3}{4}}\)
The power of the radical is the numerator of the exponent, 3. The index of the radical is the denominator of the exponent, 4.	\({(\sqrt[4]{81})}^{3}\)
Simplify.	\({(3)}^{3}\)
	\(27\)

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

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

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

The same laws of exponents that we already used apply to rational exponents, too. We will list the Exponent Properties here to have them for reference as we simplify expressions.

When we multiply the same base, we add the exponents.

Example

Try it.

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

Solution

ⓐ

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

ⓑ

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

ⓒ

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

We will use the Power Property in the next example.

Example

Try it.

Simplify: ⓐ \({({x}^{4})}^{\frac{1}{2}}\) ⓑ \({({y}^{6})}^{\frac{1}{3}}\) ⓒ \({({z}^{9})}^{\frac{2}{3}}\).

Solution

ⓐ

	\({({x}^{4})}^{\frac{1}{2}}\)
To raise a power to a power, we multiply the exponents.	\({x}^{4\cdot \frac{1}{2}}\)
Simplify.	\({x}^{2}\)

ⓑ

	\({({y}^{6})}^{\frac{1}{3}}\)
To raise a power to a power, we multiply the exponents.	\({y}^{6\cdot \frac{1}{3}}\)
Simplify.	\({y}^{2}\)

ⓒ

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

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

Key Concepts

Summary of Exponent Properties
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:
  \[\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},\ \ a\ne 0,\ \ m>n\] \[\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}},\ \ a\ne 0,\ \ n>m\]
- Zero Exponent Definition \({a}^{0}=1\), \(a\ne 0\)
- Quotient to a Power Property \({(\frac{a}{b})}^{m}=\frac{{a}^{m}}{{b}^{m}},\ b\ne 0\)

Chapter 9 Review Exercises

Simplify Expressions with Square Roots

In the following exercises, simplify.

Try it.

\(\sqrt{64}\)

Try it.

\(\sqrt{144}\)

Solution

Try it.

\(-\sqrt{25}\)

Try it.

\(-\sqrt{81}\)

Solution

\(-9\)

Try it.

\(\sqrt{-9}\)

Try it.

\(\sqrt{-36}\)

Solution

not a real number

Try it.

\(\sqrt{64}+\sqrt{225}\)

Try it.

\(\sqrt{64+225}\)

Solution

Estimate Square Roots

In the following exercises, estimate each square root between two consecutive whole numbers.

Try it.

\(\sqrt{28}\)

Try it.

\(\sqrt{155}\)

Solution

\(12<\sqrt{155}<13\)

Approximate Square Roots

In the following exercises, approximate each square root and round to two decimal places.

Try it.

\(\sqrt{15}\)

Try it.

\(\sqrt{57}\)

Solution

7.55

Simplify Variable Expressions with Square Roots

In the following exercises, simplify.

Try it.

\(\sqrt{{q}^{2}}\)

Try it.

\(\sqrt{64{b}^{2}}\)

Solution

\(8\left|b\right|\)

Try it.

\(\text{-}\sqrt{121{a}^{2}}\)

Try it.

\(\sqrt{225{m}^{2}{n}^{2}}\)

Solution

\(15\left|m\right|\left|n\right|\)

Try it.

\(\text{-}\sqrt{100{q}^{2}}\)

Try it.

\(\sqrt{49{y}^{2}}\)

Solution

\(7\left|y\right|\)

Try it.

\(\sqrt{4{a}^{2}{b}^{2}}\)

Try it.

\(\sqrt{121{c}^{2}{d}^{2}}\)

Solution

\(11\left|c\right|\left|d\right|\)

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

Try it.

\(\sqrt{300}\)

Try it.

\(\sqrt{98}\)

Solution

\(7\sqrt{2}\)

Try it.

\(\sqrt{{x}^{13}}\)

Try it.

\(\sqrt{{y}^{19}}\)

Solution

\(\left|{y}^{9}\right|\sqrt{y}\)

Try it.

\(\sqrt{16{m}^{4}}\)

Try it.

\(\sqrt{36{n}^{13}}\)

Solution

\(6{n}^{6}\sqrt{n}\)

Try it.

\(\sqrt{288{m}^{21}}\)

Try it.

\(\sqrt{150{n}^{7}}\)

Solution

\(5\left|{n}^{3}\right|\sqrt{6n}\)

Try it.

\(\sqrt{48{r}^{5}{s}^{4}}\)

Try it.

\(\sqrt{108{r}^{5}{s}^{3}}\)

Solution

\(6{r}^{2}\left|s\right|\sqrt{3rs}\)

Try it.

\(\frac{10-\sqrt{50}}{5}\)

Try it.

\(\frac{6+\sqrt{72}}{6}\)

Solution

\(1+\sqrt{2}\)

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

Try it.

\(\sqrt{\frac{16}{25}}\)

Try it.

\(\sqrt{\frac{81}{36}}\)

Solution

\(\frac{3}{2}\)

Try it.

\(\sqrt{\frac{{x}^{8}}{{x}^{4}}}\)

Try it.

\(\sqrt{\frac{{y}^{6}}{{y}^{2}}}\)

Solution

\({y}^{2}\)

Try it.

\(\sqrt{\frac{98{p}^{6}}{2{p}^{2}}}\)

Try it.

\(\sqrt{\frac{72{q}^{8}}{2{q}^{4}}}\)

Solution

\(6{q}^{2}\)

Try it.

\(\sqrt{\frac{65}{121}}\)

Try it.

\(\sqrt{\frac{26}{169}}\)

Solution

\(\frac{\sqrt{26}}{13}\)

Try it.

\(\sqrt{\frac{64{x}^{4}}{25{x}^{2}}}\)

Try it.

\(\sqrt{\frac{36{r}^{10}}{16{r}^{5}}}\)

Solution

\(\frac{3{r}^{2}\sqrt{r}}{2}\)

Try it.

\(\sqrt{\frac{48{p}^{3}{q}^{5}}{27pq}}\)

Try it.

\(\sqrt{\frac{12{r}^{5}{s}^{7}}{75{r}^{2}s}}\)

Solution

\(\frac{2\left|r\right|\left|{s}^{3}\right|\sqrt{r}}{5}\)

Condensed — the full section is in OpenStax Elementary 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.

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

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\(\frac{53}{60}\)
Simplify: \({(4{x}^{2}{y}^{5})}^{3}\).
If you missed this problem, review .

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\(64{x}^{6}{y}^{15}\)
Simplify: \({5}^{-3}\).
If you missed this problem, review .

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\(\frac{1}{125}\)
Write as a radical expression: ⓐ \({x}^{\frac{1}{2}}\) ⓑ \({y}^{\frac{1}{3}}\) ⓒ \({z}^{\frac{1}{4}}\).

답을 드러내세요

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

\({x}^{\frac{1}{2}}\)
The denominator of the 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. \(\sqrt[4]{z}\)
Write as a radical expression: ⓐ \({t}^{\frac{1}{2}}\) ⓑ \({m}^{\frac{1}{3}}\) ⓒ \({r}^{\frac{1}{4}}\).

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ⓐ \(\sqrt{t}\) ⓑ \(\sqrt[3]{m}\) ⓒ \(\sqrt[4]{r}\)
Write as a radial expression: ⓐ \({b}^{\frac{1}{2}}\) ⓑ \({z}^{\frac{1}{3}}\) ⓒ \({p}^{\frac{1}{4}}\).

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

답을 드러내세요

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

\(\sqrt{x}\)
No index is shown, so it is 2.
The denominator of the exponent will be 2. \({x}^{\frac{1}{2}}\)

ⓑ

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

ⓒ

\(\sqrt[4]{z}\)
The index is 4, so the denominator of the exponent is 4. \({z}^{\frac{1}{4}}\)
Write with a rational exponent: ⓐ \(\sqrt{s}\) ⓑ \(\sqrt[3]{x}\) ⓒ \(\sqrt[4]{b}\).

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ⓐ \({s}^{\frac{1}{2}}\) ⓑ \({x}^{\frac{1}{3}}\) ⓒ \({b}^{\frac{1}{4}}\)
Write with a rational exponent: ⓐ \(\sqrt{v}\) ⓑ \(\sqrt[3]{p}\) ⓒ \(\sqrt[4]{p}\).

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ⓐ \({v}^{\frac{1}{2}}\) ⓑ \({p}^{\frac{1}{3}}\) ⓒ \({p}^{\frac{1}{4}}\)
Write with a rational exponent: ⓐ \(\sqrt{5y}\) ⓑ \(\sqrt[3]{4x}\) ⓒ \(3\sqrt[4]{5z}\).

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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. \({(5y)}^{\frac{1}{2}}\)

ⓑ

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

ⓒ

\(3\sqrt[4]{5z}\)
The index is 4, so the denominator of the exponent is 4. \(3{(5z)}^{\frac{1}{4}}\)
Write with a rational exponent: ⓐ \(\sqrt{10m}\) ⓑ \(\sqrt[5]{3n}\) ⓒ \(3\sqrt[4]{6y}\).

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ⓐ \({(10m)}^{\frac{1}{2}}\) ⓑ \({(3n)}^{\frac{1}{5}}\) ⓒ \({(486y)}^{\frac{1}{4}}\)
Write with a rational exponent: ⓐ \(\sqrt[7]{3k}\) ⓑ \(\sqrt[4]{5j}\) ⓒ \(8\sqrt[3]{2a}\).

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ⓐ \({(3k)}^{\frac{1}{7}}\) ⓑ \({(5j)}^{\frac{1}{4}}\) ⓒ \({(1024a)}^{\frac{1}{3}}\)
Simplify: ⓐ \({25}^{\frac{1}{2}}\) ⓑ \({64}^{\frac{1}{3}}\) ⓒ \({256}^{\frac{1}{4}}\).

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ⓐ

\({25}^{\frac{1}{2}}\)
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. \(4\)

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ⓐ

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

ⓑ

	\(\text{-}{64}^{\frac{1}{3}}\)
The exponent applies only to the 64.	\(\text{-}({64}^{\frac{1}{3}})\)
Rewrite as a cube root.	\(\text{-}\sqrt[3]{64}\)
Rewrite 64 as \({4}^{3}.\)	\(\text{-}\sqrt[3]{{4}^{3}}\)
Simplify.	\(-4\)

ⓒ

	\({(64)}^{-\frac{1}{3}}\)
Rewrite as a fraction with a positive exponent, using the property, \({a}^{\text{-}n}=\frac{1}{{a}^{n}}.\)	\(\frac{1}{\sqrt[3]{64}}\)
Write as a cube root.
Rewrite 64 as \({4}^{3}.\)	\(\frac{1}{\sqrt[3]{{4}^{3}}}\)
Simplify.	\(\frac{1}{4}\)

Simplify: ⓐ \({(-125)}^{\frac{1}{3}}\) ⓑ \(\text{-}{125}^{\frac{1}{3}}\) ⓒ \({(125)}^{-\frac{1}{3}}\).

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ⓐ \(-5\) ⓑ \(-5\) ⓒ \(\frac{1}{5}\)
Simplify: ⓐ \({(-32)}^{\frac{1}{5}}\) ⓑ \(\text{-}{32}^{\frac{1}{5}}\) ⓒ \({(32)}^{-\frac{1}{5}}\).

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ⓐ \(-2\) ⓑ \(-2\) ⓒ \(\frac{1}{2}\)

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ⓐ

	\({(-16)}^{\frac{1}{4}}\)
Rewrite as a fourth root.	\(\sqrt[4]{-16}\)
There is no real number whose fourth power is \(-16.\)

ⓑ

	\(\text{-}{16}^{\frac{1}{4}}\)
The exponent only applies to the 16. Rewrite as a fourth 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.	\(\frac{1}{2}\)

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

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ⓐ not real ⓑ \(-8\) ⓒ \(\frac{1}{8}\)
Simplify: ⓐ \({(-256)}^{\frac{1}{4}}\) ⓑ \(\text{-}{256}^{\frac{1}{4}}\) ⓒ \({(256)}^{-\frac{1}{4}}\).

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ⓐ not real ⓑ \(-4\) ⓒ \(\frac{1}{4}\)
Write with a rational exponent: ⓐ \(\sqrt{{y}^{3}}\) ⓑ \(\sqrt[3]{{x}^{2}}\) ⓒ \(\sqrt[4]{{z}^{3}}\).
답을 드러내세요
We want to use \({a}^{\frac{m}{n}}=\sqrt[n]{{a}^{m}}\) to write each radical in the form \({a}^{\frac{m}{n}}\).
1. ⓐ
2. ⓑ
3. ⓒ
Write with a rational exponent: ⓐ \(\sqrt{{x}^{5}}\) ⓑ \(\sqrt[4]{{z}^{3}}\) ⓒ \(\sqrt[5]{{y}^{2}}\).

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ⓐ \({x}^{\frac{5}{2}}\) ⓑ \({z}^{\frac{3}{4}}\) ⓒ \({y}^{\frac{2}{5}}\)
Write with a rational exponent: ⓐ \(\sqrt[5]{{a}^{2}}\) ⓑ \(\sqrt[3]{{b}^{7}}\) ⓒ \(\sqrt[4]{{m}^{5}}\).

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

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ⓐ

	\({9}^{\frac{3}{2}}\)
The power of the radical is the numerator of the exponent, 3. Since the denominator of the exponent is 2, this is a square root.	\({(\sqrt{9})}^{3}\)
Simplify.	\({(3)}^{3}\)
	\(27\)

ⓑ

	\({125}^{\frac{2}{3}}\)
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.	\({(5)}^{2}\)
	\(25\)

ⓒ

	\({81}^{\frac{3}{4}}\)
The power of the radical is the numerator of the exponent, 3. The index of the radical is the denominator of the exponent, 4.	\({(\sqrt[4]{81})}^{3}\)
Simplify.	\({(3)}^{3}\)
	\(27\)

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

ⓐ

	\({16}^{-\frac{3}{2}}\)
Rewrite using \({b}^{\text{-}p}=\frac{1}{{b}^{p}}.\)	\(\frac{1}{{16}^{\frac{3}{2}}}\)
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 \({b}^{\text{-}p}=\frac{1}{{b}^{p}}.\)	\(\frac{1}{{32}^{\frac{2}{5}}}\)
Change to radical form.	\(\frac{1}{{(\sqrt[5]{32})}^{2}}\)
Rewrite the radicand as a power.	\(\frac{1}{{(\sqrt[5]{{2}^{5}})}^{2}}\)
Simplify.	\(\frac{1}{{2}^{2}}\)
	\(\frac{1}{4}\)

ⓒ

	\({4}^{-\frac{5}{2}}\)
Rewrite using \({b}^{\text{-}p}=\frac{1}{{b}^{p}}.\)	\(\frac{1}{{4}^{\frac{5}{2}}}\)
Change to radical form.	\(\frac{1}{{(\sqrt{4})}^{5}}\)
Simplify.	\(\frac{1}{{2}^{5}}\)
	\(\frac{1}{32}\)

Simplify: ⓐ \({8}^{-\frac{5}{3}}\) ⓑ \({81}^{-\frac{3}{2}}\) ⓒ \({16}^{-\frac{3}{4}}\).

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ⓐ \(\frac{1}{32}\) ⓑ \(\frac{1}{729}\) ⓒ \(\frac{1}{8}\)
Simplify: ⓐ \({4}^{-\frac{3}{2}}\) ⓑ \({27}^{-\frac{2}{3}}\) ⓒ \({625}^{-\frac{3}{4}}\).

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

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ⓐ

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

ⓑ

	\(\text{-}{25}^{-\frac{3}{2}}\)
Rewrite using \({b}^{\text{-}p}=\frac{1}{{b}^{p}}.\)	\(\text{-}(\frac{1}{{25}^{\frac{3}{2}}})\)
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 is \(-25.\)	Not a real number.

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

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ⓐ \(-64\) ⓑ \(-\frac{1}{64}\) ⓒ not a real number
Simplify: ⓐ \({-81}^{\frac{3}{2}}\) ⓑ \({-81}^{-\frac{3}{2}}\) ⓒ \({(-81)}^{-\frac{3}{2}}\).

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

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

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ⓐ

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

ⓑ

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

ⓒ

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

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

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ⓐ 9 ⓑ \({y}^{3}\) ⓒ m
Simplify: ⓐ \({5}^{\frac{3}{5}}\cdot {5}^{\frac{7}{5}}\) ⓑ \({z}^{\frac{1}{8}}\cdot {z}^{\frac{7}{8}}\) ⓒ \({n}^{\frac{2}{7}}\cdot {n}^{\frac{5}{7}}\).

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ⓐ 25 ⓑ z ⓒ n
Simplify: ⓐ \({({x}^{4})}^{\frac{1}{2}}\) ⓑ \({({y}^{6})}^{\frac{1}{3}}\) ⓒ \({({z}^{9})}^{\frac{2}{3}}\).

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ⓐ

\({({x}^{4})}^{\frac{1}{2}}\)
To raise a power to a power, we multiply the exponents. \({x}^{4\cdot \frac{1}{2}}\)
Simplify. \({x}^{2}\)

ⓑ

\({({y}^{6})}^{\frac{1}{3}}\)
To raise a power to a power, we multiply the exponents. \({y}^{6\cdot \frac{1}{3}}\)
Simplify. \({y}^{2}\)

ⓒ

\({({z}^{9})}^{\frac{2}{3}}\)
To raise a power to a power, we multiply the exponents. \({z}^{9\cdot \frac{2}{3}}\)
Simplify. \({z}^{6}\)
Simplify: ⓐ \({({p}^{10})}^{\frac{1}{5}}\) ⓑ \({({q}^{8})}^{\frac{3}{4}}\) ⓒ \({({x}^{6})}^{\frac{4}{3}}\).

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ⓐ \({p}^{2}\) ⓑ \({q}^{6}\) ⓒ \({x}^{8}\)
Simplify: ⓐ \({({r}^{6})}^{\frac{5}{3}}\) ⓑ \({({s}^{12})}^{\frac{3}{4}}\) ⓒ \({({m}^{9})}^{\frac{2}{9}}\).

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ⓐ \({r}^{10}\) ⓑ \({s}^{9}\) ⓒ \({m}^{2}\)
Simplify: ⓐ \(\frac{{x}^{\frac{4}{3}}}{{x}^{\frac{1}{3}}}\) ⓑ \(\frac{{y}^{\frac{3}{4}}}{{y}^{\frac{1}{4}}}\) ⓒ \(\frac{{z}^{\frac{2}{3}}}{{z}^{\frac{5}{3}}}\).

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ⓐ

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

ⓑ

\(\frac{{y}^{\frac{3}{4}}}{{y}^{\frac{1}{4}}}\)
To divide with the same base, we subtract the exponents. \({y}^{\frac{3}{4}-\frac{1}{4}}\)
Simplify. \({y}^{\frac{1}{2}}\)

ⓒ

\(\frac{{z}^{\frac{2}{3}}}{{z}^{\frac{5}{3}}}\)
To divide with the same base, we subtract the exponents. \({z}^{\frac{2}{3}-\frac{5}{3}}\)
Rewrite without a negative exponent. \(\frac{1}{z}\)

Symbols used here

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

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

|x|

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

\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.

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: Rational Exponents

Simplify expressions with
Simplify expressions with
Use the Laws of Exponents to simply expressions with rational exponents
If
If
For any positive integers

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 Elementary Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

에 더 Algebra

Linear equations Quadratic equations Systems of equations Inequalities Factoring Expanding Simplifying expressions Functions and graphs Exponential and logarithmic equations Polynomial equations Absolute value

	\(\sqrt{x}\)
No index is shown, so it is 2. The denominator of the exponent will be 2.	\({x}^{\frac{1}{2}}\)

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

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

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

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

	\(3\sqrt[4]{5z}\)
The index is 4, so the denominator of the exponent is 4.	\(3{(5z)}^{\frac{1}{4}}\)

	\({25}^{\frac{1}{2}}\)
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.	\(4\)

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

	\(\frac{{y}^{\frac{3}{4}}}{{y}^{\frac{1}{4}}}\)
To divide with the same base, we subtract the exponents.	\({y}^{\frac{3}{4}-\frac{1}{4}}\)
Simplify.	\({y}^{\frac{1}{2}}\)

	\(\frac{{z}^{\frac{2}{3}}}{{z}^{\frac{5}{3}}}\)
To divide with the same base, we subtract the exponents.	\({z}^{\frac{2}{3}-\frac{5}{3}}\)
Rewrite without a negative exponent.	\(\frac{1}{z}\)