Rational Exponents and Radicals

Section 2.1 Rational Exponents and Radicals

Recall that previously, powers and exponents are a concise notation to write repeated multiplication. For example, $3^4$ means multiply 3, 4 times, or $3^4 = 3 \times 3 \times 3 \times 3 = 81\text{.}$

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We can extend the definition of exponents to allow for exponents which are fractions (i.e. rational numbers). For example, expressions like,

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\begin{gather*} 3^{1/2} \quad \text{or} \quad 4^{3/2} \quad \text{or} \quad 8^{-4/3} \end{gather*}

At first, the previous definition of exponents as “how many times you’re multiplying” doesn’t seem to make sense. After all, for example, for $4^{3/2}\text{,}$ what does it mean to multiply 4 by itself “3/2” times?

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Example 2.1.1.

Suppose that we extend exponents to be allowed to be any rational number. Then, we want the laws of exponents to still work. In particular, the two fundamental laws,

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\begin{align*} \boxed{ \begin{aligned} a^m \cdot a^n \amp= a^{m+n} \\ \frac{a^m}{a^n} \amp= a^{m-n} \end{aligned} } \end{align*}

Then, consider $3^{1/2}\text{.}$ By properties of exponents, we have,

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\begin{align*} 3^{1/2} \cdot 3^{1/2} \amp = 3^{1/2 + 1/2} \\\\ \amp = 3^1 \\\\ \amp = 3 \end{align*}

This shows that $3^{1/2}$ multiplied by itself is equal to 3. In other words, $(3^{1/2})^2 = 3\text{.}$ On the other hand, we know that $\sqrt{3}$ multiplied by itself is equal to 3, or,

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\begin{gather*} \sqrt{3} \cdot \sqrt{3} = 3 \end{gather*}

This means that $3^{1/2}$ and $\sqrt{3}$ must be the same number, and so $3^{1/2} = \sqrt{3}\text{.}$

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In general, for any number $a\text{,}$ if we multiply $a^{1/2}$ by itself, we get,

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\begin{gather*} a^{1/2} \cdot a^{1/2} = a^1 = a \end{gather*}

This means that $(a^{1/2})^2 = a\text{,}$ and so $a^{1/2} = \sqrt{a}$ is the square root of $a\text{.}$

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Subsection 2.1.1 Rational Exponents

Definition 2.1.2.

\begin{gather*} \boxed{a^{1/2} = \sqrt{a}} \end{gather*}

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In other words,

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\begin{gather*} \boxed{\text{Raising a number to the power $1/2$ is equivalent to taking its square root}} \end{gather*}

Technically, there are two numbers whose square is $a\text{,}$ which are $\sqrt{a}$ and $-\sqrt{a}\text{.}$ We define $a^{1/2}$ to be the positive square root of $a\text{.}$

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In a similar way, for an exponent of $1/3\text{,}$ we have,

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\begin{align*} a^{1/3} \cdot a^{1/3} \cdot a^{1/3} \amp = a^{1/3 + 1/3 + 1/3} \\\\ \amp = a^1 \\\\ \amp = a \end{align*}

Therefore, $(a^{1/3})^3 = a\text{,}$ and so $a^{1/3} = \sqrt[3]{a}$ is the cube root of $a\text{.}$

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\begin{gather*} \boxed{\text{Raising a number to the power $1/3$ is equivalent to taking its cube root}} \end{gather*}

In general, this works for any exponent of the form $\frac{1}{n}\text{.}$

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Definition 2.1.3.

\begin{gather*} \boxed{a^{1/n} = \sqrt[n]{a}} \end{gather*}

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Next, we want to define a rational exponent when the numerator is a number other than 1, like $\frac{2}{3}\text{,}$ or $\frac{5}{2}\text{.}$

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Example 2.1.4.

Consider $8^{2/3}\text{.}$ Recognize that $\frac{2}{3}$ is just $2 \cdot \frac{1}{3}\text{,}$ so,

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\begin{align*} 8^{2/3} = 8^{2 \cdot \frac{1}{3}} \amp = (8^{1/3})^2 \amp\amp \text{by properties of exponents} \\\\ \amp = (\sqrt[3]{8})^2 \\\\ \amp = 2^2 = 4 \end{align*}

And so $8^{2/3} = (\sqrt[3]{8})^2\text{.}$ On the other hand,

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\begin{align*} 8^{2/3} = 8^{2 \cdot \frac{1}{3}} \amp = (8^2)^{1/3} \amp\amp \text{by properties of exponents} \\\\ \amp = \sqrt[3]{8^2} \\\\ \amp = \sqrt[3]{64} = 4 \end{align*}

And so $8^{2/3} = \sqrt[3]{8^2}\text{.}$ Putting both of them together, we get that all of these are equal,

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\begin{gather*} 8^{2/3} = (\sqrt[3]{8})^2 = \sqrt[3]{8^2} \end{gather*}

and all of them are equal to 4.

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In general,

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Definition 2.1.5.

\begin{gather*} \boxed{a^{m/n} = \sqrt[n]{a^m} = \brac{\sqrt[n]{a}}^m} \end{gather*}

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This means that when raising a number to a fraction power $\frac{m}{n}\text{,}$

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Take the $n$th root and then raise to the $m$th power.
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Or: Raise it to the the $m$th power and then take the $n$th root.
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In short,

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Numerator $\rightarrow$ exponent.
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Denominator $\rightarrow$ root index.
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Some notes:

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Decimal exponents can be thought of as fraction exponents. For example, $4^{0.5} = 4^{1/2} = \sqrt{4} = 2\text{.}$
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Note that when evaluating expressions with rational exponents with a calculator, be sure to put parentheses around the fraction. Most calculators will evaluate as $\frac{8^2}{3}\text{,}$ because with order of operations, exponentiation is higher than division.
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Note that this always works, except if $n$ is even and $a$ is negative, because we can’t take the square root (or any even root) of a negative number.
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We can also define negative rational exponents, in the same way as negative integer exponents.

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Definition 2.1.6.

\begin{gather*} \boxed{a^{-m/n} = \frac{1}{a^{m/n}}} \end{gather*}

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Subsection 2.1.2 Summary of Rational Exponents and Exponent Laws

In summary,

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\begin{align*} \boxed{ \begin{aligned} a^{1/n} \amp= \sqrt[n]{a} \\ a^{m/n} \amp= \sqrt[n]{a^m} = \brac{\sqrt[n]{a}}^m \\ a^{-m/n} \amp= \frac{1}{a^{m/n}} \end{aligned}} \end{align*}

All of the laws of exponents also apply to rational exponents.

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Theorem 2.1.7.

Exponent laws.

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\begin{align*} \boxed{ \begin{aligned} a^m \cdot a^n \amp= a^{m+n} \\ \frac{a^m}{a^n} \amp= a^{m-n} \\ (a^m)^n \amp= a^{mn} \\ (ab)^n \amp= a^n b^n \\ \brac{\frac{a}{b}}^n \amp= \frac{a^n}{b^n} \end{aligned} } \qquad \boxed{ \begin{aligned} a^{-n} \amp= \frac{1}{a^n} \\ \frac{1}{a^{-n}} \amp= a^n \\ \brac{\frac{a}{b}}^{-n} \amp= \brac{\frac{b}{a}}^n \end{aligned} } \end{align*}

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Subsection 2.1.3 Converting Between Rational Exponents and Radicals

Exercise Group 2.1.1.

Write each expression in exponent form.

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(a)

$\sqrt{39}$

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(b)

$\sqrt[4]{90}$

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(c)

$\sqrt[3]{29}$

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(d)

$\sqrt[5]{100}$

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(e)

$\sqrt{3^5}$

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(f)

$\brac{\sqrt[3]{25}}^2$

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(g)

$\sqrt{6^5}$

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(h)

$\brac{\sqrt[4]{19}}^3$

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Exercise Group 2.1.2.

Write each expression in radical form.

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(a)

$36^{\frac{1}{3}}$

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(b)

$48^{\frac{1}{2}}$

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(c)

$\brac{-30}^{\frac{1}{5}}$

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Exercise Group 2.1.3.

Write each expression in radical form in 2 ways.

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(a)

$40^{\frac{2}{3}}$

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(b)

$26^{\frac{2}{5}}$

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Subsection 2.1.4 Evaluating Rational Exponents Exact Values Examples

Exercise Group 2.1.4.

Evaluate each power, without using a calculator. Your final answer should be either a whole number or a rational number.

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(a)

$64^{\frac{1}{3}}$

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Answer.

$4$

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(b)

$\sqrt[5]{32}$

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Answer.

$2$

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(c)

$\brac{\frac{16}{81}}^{\frac{1}{4}}$

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Answer.

$\frac{2}{3}$

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(d)

$36^{\frac{1}{2}}$

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Answer.

$6$

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(e)

$100^{0.5}$

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Answer.

$10$

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(f)

$27^{\frac{1}{3}}$

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Answer.

$3$

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(g)

$0.25^{\frac{1}{2}}$

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Answer.

$\frac{1}{2}$

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(h)

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

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Answer.

$2$

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(i)

$16^{\frac{1}{2}}$

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Answer.

$4$

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(j)

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

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Answer.

$-2$

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(k)

$\brac{\frac{4}{9}}^{\frac{1}{2}}$

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Answer.

$\frac{2}{3}$

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(l)

$81^{0.25}$

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Answer.

$3$

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(m)

$125^{1/3}$

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Answer.

$5$

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Exercise Group 2.1.5.

Evaluate each power, without using a calculator, if possible. Your final answer should be either a whole number or a rational number.

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(a)

$27^{\frac{4}{3}}$

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Answer.

$81$

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(b)

$100^{\frac{3}{2}}$

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Answer.

$1000$

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(c)

$27^{\frac{2}{3}}$

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Answer.

$9$

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(d)

$8^{\frac{2}{3}}$

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Answer.

$4$

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(e)

$\brac{-27}^{\frac{1}{3}}$

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Answer.

$-3$

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(f)

$(-16)^{1/2}$

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Answer.

no real value

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(g)

$0.49^{\frac{1}{2}}$

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Answer.

$\frac{7}{10}$

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(h)

$\brac{-32}^{0.4}$

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Answer.

$4$

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(i)

$0.01^{\frac{3}{2}}$

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Answer.

$\frac{1}{1000}$

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(j)

$8^{\frac{4}{3}}$

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Answer.

$16$

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(k)

$0.04^{\frac{3}{2}}$

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Answer.

$\frac{1}{125}$

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(l)

$\brac{-32}^{0.2}$

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Answer.

$-2$

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(m)

$16^{0.25}$

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Answer.

$2$

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(n)

$8^{\frac{5}{3}}$

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Answer.

$32$

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(o)

$(-10000)^{1/4}$

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Answer.

no real value

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(p)

$1024^{0.2}$

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Answer.

$4$

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(q)

$\brac{-64}^{\frac{1}{3}}$

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Answer.

$-4$

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(r)

$\brac{-1000}^{\frac{1}{3}}$

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Answer.

$-10$

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(s)

$81^{\frac{3}{4}}$

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Answer.

$27$

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(t)

$\brac{-27}^{\frac{4}{3}}$

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Answer.

$81$

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(u)

$-(-1000)^{1/3}$

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Answer.

$10$

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(v)

$\brac{-27}^{-\frac{2}{3}}$

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Answer.

$\frac{1}{9}$

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(w)

$\brac{\frac{3^4}{16}}^{-0.75}$

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Answer.

$\frac{8}{27}$

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(x)

$\brac{-32}^{\frac{3}{5}}$

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Answer.

$-8$

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(y)

$\brac{81^{-0.25}}^3$

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Answer.

$\frac{1}{27}$

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Subsection 2.1.5 Rational Exponents Using a Calculator

Exercise Group 2.1.6.

Evaluate each expression using a calculator. Express your answers to four decimal places, if necessary.

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(a)

$3^{\frac{3}{4}}$

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Answer.

2.2795

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(b)

$2^{\frac{7}{8}}$

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Answer.

1.8340

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(c)

$2^{-\frac{1}{3}}$

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Answer.

0.7937

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(d)

$4^{-\frac{3}{5}}$

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Answer.

0.5743

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(e)

$\sqrt[4]{8}$

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Answer.

1.6818

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(f)

$\sqrt[5]{27}$

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Answer.

1.9332

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(g)

$\frac{1}{\sqrt[3]{7}}$

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Answer.

0.5228

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(h)

$\brac{81^{-0.25}}^3$

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Answer.

0.0370

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(i)

$\brac{\frac{2^5}{5^2}}^{-\frac{3}{2}}$

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Answer.

0.2195

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(j)

$\brac{\frac{2^3}{8^2}}^{\frac{2}{3}}$

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Answer.

0.2500

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(k)

$\brac{\frac{-64}{6^{\frac{1}{2}}}}^{\frac{4}{3}}$

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Answer.

62.5496

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(l)

$\frac{\brac{2^{\frac{1}{2}}}^3}{16}$

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Answer.

0.1768

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(m)

$1.8^{1.4}$

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Answer.

2.2771

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(n)

$0.75^{1.2}$

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Answer.

0.7081

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Subsection 2.1.6 Simplifying Algebraic Expressions with Rational Exponents

Exercise Group 2.1.7.

Simplify each expression using exponent laws. Write with rational exponents where necessary.

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(a)

$x^5 \cdot x^{-\frac{1}{2}}$

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Answer.

$x^{\frac{9}{2}}$

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(b)

$x^{1.5} \cdot x^{3.5}$

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Answer.

$x^5$

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(c)

$b^{\frac{1}{5}} \cdot b^{\frac{9}{5}}$

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Answer.

$b^2$

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(d)

$x^3 \cdot x^{-\frac{2}{3}}$

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Answer.

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

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(e)

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

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Answer.

$x^{\frac{16}{3}}$

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(f)

$p^{-\frac{5}{4}} \cdot p^{\frac{1}{2}}$

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Answer.

$\frac{1}{p^{\frac{3}{4}}}$

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(g)

$\brac{a^2}^{\frac{3}{2}}$

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Answer.

$a^3$

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(h)

$k^{4.8} \cdot k^3$

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Answer.

$k^{\frac{39}{5}}$

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(i)

$\brac{4x^3}^{0.5}$

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Answer.

$2x^{3/2}$

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(j)

$2x^{\frac{1}{3}} \cdot -4x^{\frac{5}{3}}$

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Answer.

$-8x^2$

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(k)

$\brac{9x^2}^{\frac{3}{2}}$

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Answer.

$27x^3$

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(l)

$\brac{27x^6}^{\frac{2}{3}}$

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Answer.

$9x^4$

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(m)

$(t^{\frac{4}{3}} \cdot t^{\frac{1}{3}})^9$

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Answer.

$t^{15}$

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(n)

$(x^3 \cdot x^{\frac{3}{2}})^{\frac{1}{2}}$

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Answer.

$x^{\frac{9}{4}}$

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Exercise Group 2.1.8.

Simplify each expression. Write with rational exponents where necessary.

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(a)

$(3m^{\frac{1}{2}} \cdot 4n^{\frac{2}{3}})^2$

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Answer.

$144mn^{\frac{4}{3}}$

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(b)

$(a^{\frac{1}{4}} \cdot 3b^{\frac{5}{2}})^4$

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Answer.

$81ab^{10}$

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(c)

$\brac{\dfrac{-8a^6}{27}}^{\frac{1}{3}}$

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Answer.

$-\dfrac{2a^2}{3}$

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(d)

$(9p^2)^{-\frac{1}{2}} \cdot p^{-\frac{3}{2}}$

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Answer.

$\dfrac{1}{3p^{\frac{5}{2}}}$

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(e)

$\brac{\dfrac{x^3}{64}}^{-\frac{2}{3}}$

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Answer.

$\dfrac{16}{x^2}$

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(f)

$\brac{\dfrac{m^{\frac{1}{2}} n^{\frac{1}{3}}}{n^{\frac{2}{3}} m^{\frac{1}{2}}}}^4$

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Answer.

$\dfrac{1}{n^{\frac{4}{3}}}$

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(g)

$\brac{\dfrac{x^{\frac{2}{3}}}{y^{\frac{1}{2}}}}^{-6}$

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Answer.

$\dfrac{y^3}{x^4}$

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(h)

$\dfrac{(a^2)^{-\frac{1}{4}} \cdot (a^3)^{\frac{2}{9}}}{a^2}$

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Answer.

$\dfrac{1}{a^{11/6}}$

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(i)

$\dfrac{(m^{-2})^{\frac{2}{3}}}{(m^{\frac{1}{2}})^4}$

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Answer.

$\dfrac{1}{m^{\frac{10}{3}}}$

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(j)

$\brac{\dfrac{8a^3 b^{-4}}{64a^{-9} b^2}}^{2/3}$

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Answer.

$\dfrac{a^8}{4b^4}$

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(k)

$\brac{\dfrac{4x^{-2}}{9y^{-4}}}^{-\frac{5}{2}}$

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Answer.

$\dfrac{243x^{5}}{32y^{10}}$

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(l)

$\brac{\dfrac{x^{-2}}{(xy)^4}}^{1.5}$

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Answer.

$\dfrac{1}{x^{9}y^{6}}$

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Exercise Group 2.1.9.

Simplify each expression using exponent laws. Write with rational exponents where necessary.

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(a)

$\brac{2a^{\frac{1}{2}} b^{\frac{2}{3}} c^{-1}}^3 \brac{3a^{-1} b^{\frac{1}{3}} c^{\frac{5}{2}}}^2$

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Answer.

$\dfrac{72 b^{\frac{8}{3}} c^{2}}{a^{\frac{1}{2}}}$

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(b)

$\brac{\dfrac{27x^{-3} y^{4}}{8z^{\frac{1}{2}}}}^{\frac{2}{3}}$

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Answer.

$\dfrac{9 y^{\frac{8}{3}}}{4x^{2} z^{\frac{1}{3}}}$

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(c)

$\brac{\dfrac{a^{\frac{3}{2}} b^{-\frac{5}{4}}}{c^{-\frac{1}{2}}}}^{-4}$

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Answer.

$\dfrac{b^{5}}{a^{6} c^{2}}$

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(d)

$\dfrac{\brac{x^{\frac{5}{6}} y^{-\frac{1}{3}}}^3}{x^{-\frac{1}{2}} y^{\frac{7}{6}}}$

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Answer.

$\dfrac{x^{3}}{y^{\frac{13}{6}}}$

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(e)

$\brac{16a^{4} b^{-2}}^{-\frac{3}{4}}$

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Answer.

$\dfrac{b^{\frac{3}{2}}}{8a^{3}}$

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(f)

$\brac{x^{-2} y^{3}}^{1.5} \brac{x y^{-\frac{1}{2}}}^{-3}$

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Answer.

$\dfrac{y^{6}}{x^{6}}$

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(g)

$\brac{x^{-1} y^{2} z^{-3}}^{0} \brac{49 x^{4} z^{2}}^{\frac{1}{2}}$

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Answer.

$7x^{2} z$

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(h)

$\brac{\dfrac{a^{-1} b^{2}}{c^{3}}}^{\frac{1}{2}} \brac{\dfrac{a^{3}}{b c^{-2}}}^{\frac{1}{3}}$

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Answer.

$\dfrac{a^{\frac{1}{2}} b^{\frac{2}{3}}}{c^{\frac{5}{6}}}$

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(i)

$\brac{\dfrac{64 x^{9} y^{-6}}{k^{3}}}^{-\frac{1}{3}}$

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Answer.

$\dfrac{k y^{2}}{4x^{3}}$

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(j)

$\brac{\dfrac{27 a^{6}}{125 b^{-5}}}^{\frac{2}{3}} \cdot \brac{\dfrac{5 b^{3}}{3 a^{-2}}}^{-1}$

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Answer.

$\dfrac{27 a^{2} b^{\frac{1}{3}}}{125}$

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(k)

$\dfrac{\brac{a^{2} b^{-1}}^{\frac{3}{4}}}{\brac{a^{-\frac{3}{2}} b^{\frac{5}{8}}}^{2}}$

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Answer.

$\dfrac{a^{\frac{9}{2}}}{b^{2}}$

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(l)

$\brac{\dfrac{x^{-\frac{1}{3}}}{y^{\frac{1}{2}} z^{-2}}}^{-6}$

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Answer.

$\dfrac{x^{2} y^{3}}{z^{12}}$

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(m)

$\brac{16 a^{-4} b^{6} c^{-2}}^{\frac{1}{4}} \brac{a^{3} b^{-1} c^{5}}^{\frac{1}{2}}$

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Answer.

$2 a^{\frac{1}{2}} b c^{2}$

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(n)

$\brac{\dfrac{9x^{4}}{y^{2}}}^{-1} \brac{\dfrac{3x^{-2}}{y^{-1}}}^2$

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Answer.

$\dfrac{y^{4}}{x^{8}}$

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(o)

$\brac{a^{\frac{1}{2}} b^{\frac{1}{3}} c^{\frac{1}{4}}}^{12}$

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Answer.

$a^{6} b^{4} c^{3}$

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(p)

$\brac{\dfrac{x^{2}}{4y^{-1}}}^{-\frac{3}{2}}$

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Answer.

$\dfrac{8}{x^{3} y^{\frac{3}{2}}}$

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(q)

$\brac{\dfrac{32 m^{-3} n^{5}}{2 p^{4}}}^{\frac{1}{5}} \brac{m^{2} p^{-3}}^{\frac{3}{2}}$

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Answer.

$\dfrac{2^{\frac{4}{5}} m^{\frac{12}{5}} n}{p^{\frac{53}{10}}}$

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(r)

$\brac{\dfrac{a^{\frac{1}{2}} b^{-2}}{c^{-\frac{3}{2}}}}^{2} \brac{\dfrac{c}{a^{-1} b^{3}}}^{-1}$

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Answer.

$\dfrac{c^{2}}{b}$

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(s)

$\brac{\dfrac{x y^{-1}}{z^{2}}}^{3} \brac{\dfrac{z^{-1}}{x^{-2} y}}^{2}$

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Answer.

$\dfrac{x^{7}}{y^{5} z^{8}}$

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(t)

$\dfrac{\brac{81 a^{-4} b^{2}}^{\frac{3}{4}}}{\brac{3 a b^{-1}}^{2}}$

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Answer.

$\dfrac{3 b^{\frac{7}{2}}}{a^{5}}$

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(u)

$\brac{\dfrac{64}{x^{-6} y^{3}}}^{-\frac{2}{3}}$

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Answer.

$\dfrac{y^{2}}{16 x^{4}}$

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(v)

$\brac{\dfrac{a^{-\frac{2}{3}} b^{\frac{5}{6}}}{c^{-\frac{1}{2}}}}^{3} \brac{\dfrac{c^{2}}{a b^{-\frac{1}{3}}}}^{\frac{1}{2}}$

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Answer.

$\dfrac{b^{\frac{8}{3}} c^{\frac{5}{2}}}{a^{\frac{5}{2}}}$

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(w)

$\brac{\dfrac{x^{\tfrac{3}{2}}}{3 y^{-\tfrac{9}{2}}}}^{-2} \cdot \brac{9 x^{-1} y^{2}}^{\tfrac{1}{2}}$

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Answer.

$\dfrac{27}{x^{\frac{7}{2}} y^{8}}$

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(x)

$\brac{\dfrac{2 a^{3} b^{-1}}{5 c^{2}}}^{-3} \brac{\dfrac{25 c}{4 a^{-2} b^{\frac{1}{2}}}}^{\frac{1}{2}}$

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Answer.

$\dfrac{625 b^{\frac{11}{4}} c^{\frac{13}{2}}}{16 a^{8}}$

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(y)

$\brac{\dfrac{a b^{-1}}{c^{\frac{1}{2}}}}^{\frac{1}{3}} \brac{\dfrac{c^{-2}}{a^{-1} b}}^{-2}$

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Answer.

$\dfrac{b^{\frac{5}{3}} c^{\frac{23}{6}}}{a^{\frac{5}{3}}}$

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