B1.2 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

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

Official curriculum reference

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B1.2 · Evaluate numerical expressions with integer and rational exponents

Ontario Grade 11 Mathematics

Use exponent rules to find exact numerical values

What you will learn

1. Prerequisite bridge: powers and order of operations

2. Integer exponents: zero and negative powers

3. Rational exponents: roots and powers

Exponent meaning at a glance

Combine integer and rational exponents

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MCF3M

About this lesson