Radicals and rational exponents | Lesson (article)

What are radicals and rational exponents?

Exponential expressions are algebraic expressions with a coefficient, one or more variables, and one or more exponents. For example, in the expression $3 x^{4}$ ‍:

$3$ ‍ is the coefficient.
$x$ ‍ is the base.
$4$ ‍ is the exponent.

In $3 x^{4}$ ‍, $3$ ‍ is multiplied by $x$ ‍ $4$ ‍ times:

$3 x^{4} = 3 \cdot (x \cdot x \cdot x \cdot x)$ ‍

An expression can also be raised to an exponent. For example, for $(3 x)^{4}$ ‍, the expression $3 x$ ‍ is multiplied by itself $4$ ‍ times:

$(3 x)^{4} = 3 x \cdot 3 x \cdot 3 x \cdot 3 x = 81 x^{4}$ ‍

Notice how $3 x^{4} \neq (3 x)^{4}$ ‍ !

Rational exponents refer to exponents that are/can be represented as fractions: $\frac{1}{2}$ ‍, $3$ ‍, and $- \frac{2}{3}$ ‍ are all considered rational exponents. Radicals are another way to write rational exponents. For example, $x^{^{\frac{1}{2}}}$ ‍ and $\sqrt{x}$ ‍ are equivalent.

In this lesson, we'll:

Review the rules of exponent operations with integer exponents
Apply the rules of exponent operations to rational exponents
Make connections between equivalent rational and radical expressions

You can learn anything. Let's do this!

What are the rules of exponent operations?

Powers of products & quotients (integer exponents)

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Powers of products & quotients (integer exponents)

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The rules of exponent operations

Adding and subtracting exponential expressions

When adding and subtracting exponential expressions, we're essentially combining like terms. That means we can only combine exponential expressions with both the same base and the same exponent.

$\begin{aligned} a x^{n} \pm b x^{n} & = (a \pm b) x^{n} \end{aligned}$ ‍

$\begin{aligned} 2 x^{2} + x^{2} & = (2 + 1) x^{2} \\ = 3 x^{2} \end{aligned}$ ‍

$\begin{aligned} 5 x^{3} - 4 x^{3} & = (5 - 4) x^{3} \\ = x^{3} \end{aligned}$ ‍

$4 x^{3}$ ‍ and $4 x^{2}$ ‍ cannot be combined because the two terms have the same base, but not the same exponent.

$2 x^{2}$ ‍ and $2 y^{2}$ ‍ cannot be combined because the two terms have the same exponent, but not the same base.

Multiplying and dividing exponential expressions

When multiplying two exponential expressions with the same base, we keep the base the same, multiply the coefficients, and add the exponents. Similarly, when dividing two exponential expressions with the same base, we keep the base the same and subtract the exponents.

$\begin{aligned} a x^{m} \cdot b x^{n} & = a b \cdot x^{m + n} \\ \frac{a x^{m}}{b x^{n}} & = \frac{a}{b} \cdot x^{m - n} \end{aligned}$ ‍

$\begin{aligned} 3 x \cdot 4 x^{2} & = (3 \cdot 4) \cdot x^{1 + 2} \\ = 12 x^{3} \end{aligned}$ ‍

$\begin{aligned} \frac{10 x^{3}}{5 x^{2}} & = \frac{10}{5} \cdot x^{3 - 2} \\ = 2 x \end{aligned}$ ‍

When multiplying or dividing exponential expressions with the same exponent but different bases, we multiply or divide the bases and keep the exponents the same.

$\begin{aligned} x^{n} \cdot y^{n} & = (x y)^{n} \\ \frac{x^{n}}{y^{n}} & = {(\frac{x}{y})}^{n} \end{aligned}$ ‍

$\begin{aligned} 3^{2} \cdot 4^{2} & = (3 \cdot 4)^{2} \\ = 12^{2} \\ = 144 \end{aligned}$ ‍

$\begin{aligned} \frac{6^{4}}{2^{4}} & = {(\frac{6}{2})}^{4} \\ = 3^{4} \\ = 81 \end{aligned}$ ‍

Raising an exponential expression to an exponent and change of base

When raising an exponential expression to an exponent, raise the coefficient of the expression to the exponent, keep the base the same, and multiply the two exponents.

${(a x^{m})}^{n} = a^{n} \cdot x^{m n}$ ‍

$\begin{aligned} (x^{2})^{3} & = x^{2 \cdot 3} \\ = x^{6} \end{aligned}$ ‍

Negative exponents

A base raised to a negative exponent is equivalent to $1$ ‍ divided by the base raised to the

of the exponent.

$x^{- n} = \frac{1}{x^{n}}$ ‍

$\begin{aligned} 2 x^{- 3} & = 2 \cdot \frac{1}{x^{3}} \\ = \frac{2}{x^{3}} \end{aligned}$ ‍

$\begin{aligned} \frac{x^{5}}{x^{7}} & = x^{5 - 7} \\ = x^{- 2} \\ = \frac{1}{x^{2}} \end{aligned}$ ‍

$\begin{aligned} \frac{x^{- 3}}{y^{- 4}} & = \frac{1}{x^{3}} \div \frac{1}{y^{4}} \\ = \frac{1}{x^{3}} \cdot \frac{y^{4}}{1} \\ = \frac{y^{4}}{x^{3}} \end{aligned}$ ‍

Zero exponent

A nonzero base raised to an exponent of $0$ ‍ is equal to $1$ ‍.

$x^{0} = 1, x \neq 0$ ‍

How do the rules of exponent operations apply to rational exponents?

Every rule that applies to integer exponents also applies to rational exponents.

Try it!

try: divide two rational expressions

In order to divide $12 x^{^{\frac{5}{2}}}$ ‍ by $3 x^{^{\frac{1}{2}}}$ ‍, we

the coefficients and

the exponents of $x$ ‍.

$\frac{12 x^{^{\frac{5}{2}}}}{3 x^{^{\frac{1}{2}}}} =$ ‍

Try: raise an exponential expression to an exponent

To calculate ${(2 y^{^{\frac{4}{3}}})}^{3}$ ‍, we

and

the exponents $\frac{4}{3}$ ‍ and $3$ ‍.

${(2 y^{^{\frac{4}{3}}})}^{3} =$ ‍

How are radicals and fractional exponents related?

Rewriting roots as rational exponents

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Rewriting roots as rational exponents

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Roots and rational exponents

Squares and square roots are inverse operations: they "undo" each other. For example, if we take the square root of $3$ ‍ squared, we get $\sqrt{3^{2}} = 3$ ‍.

The reason for this becomes more apparent when we rewrite square root as a fractional exponent: $\sqrt{x} = x^{^{\frac{1}{2}}}$ ‍, and $\sqrt{3^{2}} = (3^{2})^{^{\frac{1}{2}}} = 3^{1}$ ‍.

Try it!

Try: determine equivalent expressions

Determine whether each of the radical expressions below is equivalent to $x^{^{\frac{3}{2}}} y^{^{\frac{1}{3}}}$ ‍.

	Equivalent	Not equivalent
$\sqrt{x^{3}} \cdot \sqrt[3]{y}$ ‍
$\sqrt{x y}$ ‍
$\sqrt{x^{3} y}$ ‍
$\sqrt[6]{x^{9} y^{2}}$ ‍

Your turn!

Practice: multiply rational expressions

Which of the following is equivalent to $2 x^{3} \cdot 3 x^{5}$ ‍ ?

Choose 1 answer:

(Choice A)
$5 x^{8}$ ‍
(Choice B)
$6 x^{8}$ ‍
(Choice C)
$6 x^{15}$ ‍
(Choice D)
$8 x^{15}$ ‍

Practice: change bases

If $a^{^{\frac{b}{2}}} = 25$ ‍ for positive integers $a$ ‍ and $b$ ‍, what is one possible value of $b$ ‍ ?

Practice: raise to a negative exponent

If $n^{^{- \frac{1}{3}}} = x$ ‍, where $n > 0$ ‍, what is $n$ ‍ in terms of $x$ ‍ ?

Choose 1 answer:

(Choice A)
$- \frac{1}{x^{3}}$ ‍
(Choice B)
$\frac{1}{x^{3}}$ ‍
(Choice C)
$- \sqrt[3]{x}$ ‍
(Choice D)
$\sqrt[3]{x}$ ‍

Practice: simplify radical expressions

$\frac{\sqrt[3]{8 x^{8} y^{6}}}{\sqrt{4 x^{2} y^{6}}}$ ‍

Which of the following is equivalent to the expression above?

Choose 1 answer:

(Choice A)
$x$ ‍
(Choice B)
$\frac{x \sqrt[3]{x^{2}}}{y}$ ‍
(Choice C)
$\sqrt[3]{2} x^{2}$ ‍
(Choice D)
$\sqrt{2} x^{3}$ ‍

Things to remember

Adding and subtracting exponential expressions:

$\begin{aligned} a x^{n} \pm b x^{n} & = (a \pm b) x^{n} \end{aligned}$ ‍

Multiplying and dividing exponential expressions:

$\begin{aligned} a x^{m} \cdot b x^{n} & = a b \cdot x^{m + n} \\ \frac{a x^{m}}{b x^{n}} & = \frac{a}{b} \cdot x^{m - n} \\ x^{n} \cdot y^{n} & = (x y)^{n} \\ \frac{x^{n}}{y^{n}} & = {(\frac{x}{y})}^{n} \end{aligned}$ ‍

Raising an exponential expression to an exponent and change of base:

$\begin{aligned} {(a x^{m})}^{n} & = a^{n} \cdot x^{m n} \\ (a^{b})^{n} & = a^{b n} \end{aligned}$ ‍

Negative exponent:

$x^{- n} = \frac{1}{x^{n}}$ ‍

Zero exponent:

$x^{0} = 1, x \neq 0$ ‍

All of the rules that apply to exponential expressions with integer exponents also apply to exponential expressions with fractional exponents.

$\begin{aligned} \sqrt[n]{x^{m}} & = x^{^{\frac{m}{n}}} \\ \sqrt[n]{x} \cdot \sqrt[n]{y} & = \sqrt[n]{x y} \\ \frac{\sqrt[n]{x}}{\sqrt[n]{y}} & = \sqrt[n]{\frac{x}{y}} \end{aligned}$ ‍

Radicals and rational exponents | Lesson (article) | Khan Academy (2024)

What are radicals and rational exponents?

What are the rules of exponent operations?

Powers of products & quotients (integer exponents)

The rules of exponent operations

Adding and subtracting exponential expressions

Multiplying and dividing exponential expressions

Raising an exponential expression to an exponent and change of base

Negative exponents

Zero exponent

How do the rules of exponent operations apply to rational exponents?

Try it!

How are radicals and fractional exponents related?

Rewriting roots as rational exponents

Roots and rational exponents

Try it!

Your turn!

Things to remember