Evaluate numerical expressions with integer and rational exponents

Ontario Grade 11 Mathematics

Exponential Functions

Use exponent rules to find exact numerical values

An exponent tells how a base is used in a power. For example,

3^4

means multiply four factors of

3

. Some expressions use exponents that are zero, negative, or fractions. These follow connected rules. In this lesson, you will use those rules to evaluate numerical expressions. A numerical expression contains numbers and operations but no unknown variable to solve for.

What you will learn

Explain what zero, negative, and rational exponents mean for a numerical base.
Evaluate numerical expressions by applying exponent rules and order of operations.
Recognize how parentheses and the position of a negative sign affect a value.
Check whether an exact answer is reasonable.

1. Prerequisite bridge: powers and order of operations

A base is the number being raised to a power. An exponent tells how many times the base is used as a factor when the exponent is a positive whole number. Thus,

4^3

means

4\times4\times4

, which equals

64

.

When an expression has several operations, use order of operations: evaluate powers before multiplication, division, addition, and subtraction. Parentheses show which parts belong together. For example,

(-2)^2

means

(-2)\times(-2)

, while

-2^2

means the opposite of

2^2

. The first value is

4

; the second is

-4

.

A rational number is a number that can be written as a fraction of integers, such as

\frac{9}{16}

. A rational exponent is a fraction used as an exponent, such as

\frac{1}{2}

or

\frac{3}{2}

. We will connect these exponents to roots and powers.

a^n=a\times a\times\cdots\times a

For a positive whole-number exponent, multiply the base by itself the stated number of times.
Evaluate powers before carrying out multiplication, division, addition, or subtraction.
Parentheses determine whether a negative sign is part of the base.

2. Integer exponents: zero and negative powers

An integer is a whole number, its negative, or zero. For a nonzero base, an exponent of zero gives a value of

1

. This rule fits the way powers with the same base divide: each time the exponent drops by one, the value is divided by the base. For example,

5^2=25

,

5^1=5

, and

5^0=1

.

A negative exponent means to take the reciprocal of the corresponding positive power. A reciprocal is the value that makes a product of

1

with the original number. The reciprocal of

3

is

\frac{1}{3}

. So

3^{-2}=\frac{1}{3^2}=\frac{1}{9}

. The base cannot be zero in this rule, because zero has no reciprocal.

For a fraction raised to a negative exponent, first invert the fraction and then use the positive exponent. Keep track of the entire base using parentheses. For instance,

\left(\frac{2}{5}\right)^{-1}=\frac{5}{2}

. The negative exponent does not make the answer negative; it indicates a reciprocal.

a^{-n}=\frac{1}{a^n},\quad a\ne0

For $a\ne0$ , $a^0=1$ .
For $a\ne0$ and positive integer $n$ , $a^{-n}=\frac{1}{a^n}$ .
A negative exponent changes a nonzero base to its reciprocal; it does not change its sign by itself.

3. Rational exponents: roots and powers

A square root is a number that, when squared, gives the number under the root. For example,

\sqrt{25}=5

because

5^2=25

. For positive numbers, an exponent of

\frac{1}{2}

means square root. More generally, an exponent of

\frac{1}{m}

means the

m

th root, where

m

is a positive integer.

The numerator of a rational exponent tells the power. The denominator tells the root. For example,

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

means take the cube root of

8

, then square the result:

2^2=4

. This order is useful when the root is easy to find first.

A rational exponent can also be read in the other order: raise the base to the numerator, then take the root given by the denominator. Choose the order that makes the arithmetic simpler. For negative bases, an even root of a negative number is not a real number. Parentheses also matter:

(-3)^2

is positive, but

-3^2

is negative.

a^{\frac{m}{n}}=\left(\sqrt[n]{a}\right)^m

The denominator of the exponent gives the root; the numerator gives the power.
When possible, take a simple root first to keep the numbers manageable.
Check the base and parentheses before evaluating a fractional power.

Exponent meaning at a glance

Exponent form	Meaning	Example value
Positive integer	Repeated multiplication	$3^2=9$
Zero	Value is $1$ for a nonzero base	$7^0=1$
Negative integer	Reciprocal of the positive power	$2^{-3}=\frac{1}{8}$
Rational exponent	Root from denominator, power from numerator	$16^{\frac{3}{4}}=8$

Worked example

Combine integer and rational exponents

Evaluate

\left(\frac{16}{81}\right)^{\frac{3}{4}}+2^{-2}\times\sqrt{49}

.

Read the expression
The expression has a rational exponent, a negative integer exponent, a square root, and multiplication. Evaluate the powers and root before adding. In the first term, the denominator $4$ tells us to take a fourth root, and the numerator $3$ tells us to cube the result.
Evaluate the rational power
The fourth root of $\frac{16}{81}$ is $\frac{2}{3}$ because $2^4=16$ and $3^4=81$ . Now cube that fraction.
$\left(\frac{16}{81}\right)^{\frac{3}{4}}=\left(\frac{2}{3}\right)^3=\frac{8}{27}$
Evaluate the negative power and root
A negative exponent means take the reciprocal of the positive power. Also, the square root of $49$ is $7$ .
$2^{-2}=\frac{1}{2^2}=\frac{1}{4},\quad\sqrt{49}=7$
Multiply, then add
Multiplication comes before addition. The product is $\frac{7}{4}$ . To add it to $\frac{8}{27}$ , use the common denominator $108$ .
$\frac{1}{4}\times7=\frac{7}{4},\quad\frac{8}{27}+\frac{7}{4}=\frac{32}{108}+\frac{189}{108}=\frac{221}{108}$

Answer:

\frac{221}{108}

Check: The first term is less than

1

, and the second term is

\frac{7}{4}=1.75

. Their sum is a little above

2

, consistent with

\frac{221}{108}\approx2.05

.

Common mistakes and how to avoid them

Treating a negative exponent as a negative answer.

Correction: A negative exponent means reciprocal. For example,

4^{-1}=\frac{1}{4}

, which is positive.

Reading

-3^2

as

(-3)^2

.

Correction: Without parentheses, evaluate the power first and then apply the negative sign:

-3^2=-(3^2)=-9

. With parentheses,

(-3)^2=9

.

Applying the numerator of a rational exponent as the root and the denominator as the power.

Correction: The denominator gives the root and the numerator gives the power. For example,

27^{\frac{2}{3}}=(\sqrt[3]{27})^2=9

.

Adding terms before evaluating their powers or products.

Correction: Follow order of operations. Evaluate roots and powers first, then multiplication or division, and then addition or subtraction.

Lesson summary

Positive integer exponents represent repeated multiplication.
A nonzero base raised to zero equals $1$ .
A negative exponent means the reciprocal of the corresponding positive power.
For a rational exponent, use the denominator for the root and the numerator for the power.
Use parentheses carefully and follow order of operations.

Check your understanding

Question 1

What is

5^{-2}

?

$-25$
$\frac{1}{25}$
$25$
$-\frac{1}{25}$

Show answer and explanation

\frac{1}{25}

A negative exponent means reciprocal:

5^{-2}=\frac{1}{5^2}=\frac{1}{25}

.

Question 2

Evaluate

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

.

$9$
$27$
$\frac{1}{27}$
$64$

Show answer and explanation

27

The fourth root of

81

is

3

. Then cube:

3^3=27

.

Question 3

Evaluate

-4^2+(-4)^2

.

$0$
$16$
$32$
$-32$

Show answer and explanation

0

The first term is

-(4^2)=-16

. The second is

(-4)^2=16

. Their sum is

0

.

Key terms

Base: The number raised to an exponent.
Exponent: The number that tells how a base is used in a power.
Integer: A whole number, its negative, or zero.
Rational number: A number that can be written as a fraction of integers, with a nonzero denominator.
Rational exponent: An exponent written as a fraction; its denominator indicates a root and its numerator indicates a power.
Reciprocal: For a nonzero number, the value that gives a product of $1$ when multiplied by that number.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation B1.2. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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