Interpret powers with rational exponents

Ontario Grade 11 Mathematics

Exponential Functions

Ontario Grade 11 Mathematics · B1.1

An exponent does not have to be a whole number. A fraction used as an exponent is called a rational exponent. It connects two familiar operations: powers and roots. For example, a square root asks which number, when squared, gives the original number. A rational exponent packages that root relationship into exponent notation. In this lesson, you will read that notation in words, connect it to roots, and evaluate examples by using familiar whole-number powers.

What you will learn

Explain what the numerator and denominator of a rational exponent represent.
Rewrite powers with rational exponents using roots and whole-number powers.
Interpret negative rational exponents as reciprocals.
Check whether a real-number expression is defined before evaluating it.

1. Prerequisite bridge: powers and roots

A power is a compact way to show repeated multiplication. In

5^3

, the base is

5

and the exponent is

3

. It means

5\times5\times5

, which equals

125

. The exponent tells how many equal factors of the base are multiplied.

A root reverses a power. The square root of

25

is

5

because

5^2=25

. The cube root of

8

is

2

because

2^3=8

. The small number that tells which root to take is called the root index. A square root has index

2

, though that index is usually not written.

These familiar ideas are enough to begin interpreting rational exponents. You do not need a new kind of multiplication. You need to recognize that the denominator of the exponent names a root, while the numerator names a power.

a^n=\underbrace{a\times a\times\cdots\times a}_{n\text{ factors}}

A whole-number exponent counts repeated factors.
A root asks which number gives the original value when raised to a power.
The square-root index is understood to be $2$ .

2. Reading the numerator and denominator

A rational exponent is an exponent written as a fraction. For a positive base, the denominator tells you the root to take, and the numerator tells you the power to apply. You can take the root first or take the power first when both operations give real values.

For example, an exponent of

\frac{1}{2}

means take a square root. An exponent of

\frac{1}{3}

means take a cube root. An exponent of

\frac{2}{3}

means take a cube root and then square the result, or square first and then take a cube root.

This gives two useful equivalent readings. The expression

a^{m/n}

can mean the

n

th root of

a^m

, or the

m

th power of the

n

th root of

a

. The fraction must be in simplest form when you use its denominator to name the root.

For real-number work, an even root such as a square root requires a nonnegative value under the root. An odd root, such as a cube root, can be taken for negative values too. This domain check prevents a real-valued expression from being treated as though it always has a real answer.

a^{m/n}=\sqrt[n]{a^m}=(\sqrt[n]{a})^m

The denominator names the root index.
The numerator names the whole-number power.
Reduce the exponent fraction before using its denominator as the root index.
Even roots of negative numbers are not real; odd roots of negative numbers are real.

3. A visual map and the role of a negative exponent

The table links common fractional exponents to root language. The examples use positive bases so that each root shown is real. Read across a row to see how the compact exponent notation expands into familiar operations.

A negative exponent adds one more idea: it means take the reciprocal. A reciprocal is the value that gives

1

when multiplied by the original nonzero value. Thus, a negative rational exponent means first interpret the positive fractional exponent, then take its reciprocal. The base cannot be zero because zero has no reciprocal.

For instance, the meaning of

16^{-1/2}

is the reciprocal of the square root of

16

. Since that root is

4

, the expression is

\frac{1}{4}

. The minus sign does not make the answer negative; it signals a reciprocal.

a^{-m/n}=\frac{1}{a^{m/n}}

A negative exponent means reciprocal, not a negative answer.
A base of zero is not allowed with a negative exponent.
Interpret the fraction in the exponent before taking its reciprocal.

4. Using the meaning to evaluate and explain

To interpret an expression, name the root and power before calculating. Then evaluate the root or power using a familiar whole-number fact. For a negative rational exponent, find the positive-exponent value first and then take its reciprocal.

A good written explanation is more than a final number. It states what the exponent means. For example, saying that an exponent has denominator

3

tells the reader that a cube root is involved. This interpretation is useful even when a root is not a whole number: the exponent still describes the operation, whether or not the result is easy to calculate mentally.

Before evaluating, check the base and the root index. If the reduced denominator is even, a negative base does not produce a real root. If the denominator is odd, the root of a negative base can be real. Also check for a zero base when the exponent is negative.

a^{-m/n}=\frac{1}{\sqrt[n]{a^m}}

Translate the fraction into root and power language before evaluating.
Use known whole-number powers to identify exact roots.
Check real-number restrictions, especially for even roots and negative exponents.

Common readings of rational exponents

Exponent	Root-and-power reading	Example value
$a^{1/2}$	Square root of $a$	$25^{1/2}=5$
$a^{1/3}$	Cube root of $a$	$8^{1/3}=2$
$a^{2/3}$	Cube root of $a$ , then square	$8^{2/3}=4$
$a^{-1/2}$	Reciprocal of the square root of $a$	$16^{-1/2}=\frac{1}{4}$

Worked example

Interpret a positive rational exponent

Evaluate

64^{2/3}

and explain what the exponent tells you to do.

Read the exponent
The denominator is $3$ , so a cube root is involved. The numerator is $2$ , so the result is squared. The base is positive, so the cube root is a real number.
$64^{2/3}=(\sqrt[3]{64})^2$
Find the root
The cube root of $64$ is $4$ , because $4^3=64$ .
$\sqrt[3]{64}=4$
Apply the numerator
Square the root value. This gives the value of the original expression.
$4^2=16$

Answer:

64^{2/3}=16

. The exponent means take the cube root of

64

, then square it.

Check: The equivalent order also works: square

64

to get

4096

, then take its cube root, which is

16

because

16^3=4096

.

Common mistakes and how to avoid them

Reading

a^{2/3}

as the square root of

a^3

.

Correction: The denominator gives the root index and the numerator gives the power: take the cube root and square, or square and then take the cube root.

Treating a negative exponent as a direction to make the answer negative.

Correction: A negative exponent means take a reciprocal. For example,

16^{-1/2}=\frac{1}{4}

, not

-4

.

Using an even root of a negative number as a real number.

Correction: In the real numbers, an even root needs a nonnegative value under the root. Check the base before evaluating.

Using the denominator before reducing the exponent fraction.

Correction: Write the fraction in simplest form first. Then its denominator gives the root index.

Lesson summary

A rational exponent connects powers and roots.
In $a^{m/n}$ , the denominator names the root and the numerator names the power.
In $a^{-m/n}$ , the negative sign means take the reciprocal of $a^{m/n}$ .
Check whether the requested root is real, and do not take a reciprocal of zero.

Check your understanding

Question 1

Which operation does

81^{1/2}

represent?

The square root of $81$
The cube root of $81$
The reciprocal of $81$
correctIndex ות

Show answer and explanation

The square root of

81

The denominator is

2

, so the exponent represents a square root.

Question 2

What is

125^{2/3}

?

$25$
$5$
$625$
correctIndex-offsetof

Show answer and explanation

25

The cube root of

125

is

5

, and squaring gives

25

.

Question 3

What is

9^{-1/2}

?

$3$
$-3$
$\frac{1}{3}$
correctIndex

Show answer and explanation

\frac{1}{3}

The square root of

9

is

3

. The negative exponent means take its reciprocal, giving

\frac{1}{3}

.

Key terms

Base: The number being raised to a power.
Exponent: The number that tells how a base is used in a power.
Rational exponent: An exponent written as a fraction.
Root index: The number that identifies which root is taken, such as $3$ for a cube root.
Reciprocal: A nonzero value's multiplicative inverse; multiplying the value by its reciprocal gives $1$ .

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