Evaluate numerical expressions with integer exponents

Ontario Grade 11 Mathematics

Mathematical Models

Ontario Grade 11 MBF3C — A2.2

An exponent tells how a base is used in a repeated multiplication. For example, a positive exponent such as

3

means multiply the base by itself three times. Integer exponents also include zero and negative whole numbers. To evaluate an expression correctly, first interpret each exponent, then use the order of operations. This lesson reviews the needed multiplication and fraction ideas before using them in numerical expressions.

What you will learn

Explain what a positive, zero, or negative integer exponent means.
Evaluate numerical expressions with integer exponents.
Apply the order of operations and handle negative bases carefully.

1. Review: bases, powers, and order of operations

A power is an expression made from a base and an exponent. The base is the number being raised to a power. The exponent tells how the base is used. In

5^3

, the base is

5

and the exponent is

3

. The expression means

5 \times 5 \times 5

, not

5 \times 3

.

A negative number can be the base. Parentheses make this clear:

(-2)^3

means multiply

-2

by itself three times. Without parentheses,

-2^3

means take the power first and then apply the negative sign. These expressions can have different values.

When an expression has several operations, use the order of operations: evaluate powers first, then multiplication and division from left to right, then addition and subtraction from left to right. Parentheses show which part to evaluate together. A fraction bar also groups the entire numerator and denominator.

a^n=\underbrace{a× a×\cdots× a}_{n\text{ factors}}, n>0

A power such as $4^3$ means $4 \times 4 \times 4$ .
Use parentheses when a negative number is the base.
Evaluate powers before multiplication, division, addition, or subtraction.

2. Zero and negative integer exponents

A zero exponent gives a value of

1

when the base is not zero. For example,

7^0=1

. This rule applies to negative and fractional bases as well, as long as the base is not zero.

A negative exponent means take the reciprocal of the base raised to the matching positive exponent. The reciprocal of a non-zero number is the number that multiplies it to make

1

. For example, the reciprocal of

4

is

\frac{1}{4}

, and the reciprocal of

\frac{2}{3}

is

\frac{3}{2}

.

So

4^{-2}

means

\frac{1}{4^2}

, which is

\frac{1}{16}

. A negative exponent does not make the value negative. The sign depends on the base and on how many factors are multiplied. A base of zero cannot have a zero or negative exponent because taking its reciprocal would require division by zero.

a^0=1, a^{-n}=\frac{1}{a^n} (a\ne0,\ n>0)

A non-zero base raised to zero equals $1$ .
A negative exponent means use the reciprocal, then evaluate the positive power.
A negative exponent does not by itself make an answer negative.

3. A reliable method for evaluating

Start by identifying each base and exponent. Rewrite negative powers as reciprocals so that the exponent is positive. Evaluate powers, paying attention to parentheses and signs. Then complete multiplication and division from left to right, followed by addition and subtraction from left to right.

Keep fractions exact when possible. For example, it is usually clearer to keep

\frac{1}{9}

than to turn it into a rounded decimal. If an expression contains a negative base in parentheses, an even number of identical negative factors gives a positive product, while an odd number gives a negative product.

Do not combine terms just because they look similar. The goal here is to evaluate the numerical expression as written. Follow the operation order and keep each step connected to the original expression.

Rewrite negative exponents as reciprocals before calculating.
Follow the order of operations after interpreting the exponents.
Check whether parentheses include the negative sign in the base.

4. Guided practice and application

The first example combines a negative exponent with multiplication and addition. It shows why the exponent must be handled before the other operations. The second example uses a negative base and parentheses, so the sign of the power matters.

In practical settings, powers can describe repeated multiplication, such as a quantity multiplied by the same factor several times. A numerical expression may also include negative or zero exponents as part of a calculation. The evaluation process stays the same: interpret each power, then use the order of operations.

A numerical expression can include powers, fractions, and ordinary operations.
Calculate each power correctly before moving to the remaining operations.

Integer exponent meanings

Exponent	Meaning	Example
Positive	Multiply the base by itself the indicated number of times	$3^2=3\times3=9$
Zero	A non-zero base raised to zero is $1$	$6^0=1$
Negative	Use the reciprocal of the corresponding positive power	$2^{-3}=\frac{1}{2^3}=\frac{1}{8}$

Worked example

A negative exponent in an expression

Evaluate

18-2^{-2}\times 8

.

Rewrite the negative exponent
The base is $2$ , and the exponent is $-2$ . Replace this power with the reciprocal of $2^2$ . This gives a positive fraction, not a negative value.
$2^{-2}=\frac{1}{2^2}=\frac{1}{4}$
Multiply before subtracting
The order of operations says to multiply before subtracting. Multiply $\frac{1}{4}$ by $8$ to get $2$ .
$\frac{1}{4}× 8=2$
Subtract
Now subtract the product from $18$ .
$18-2=16$

Answer:

16

Check: Since

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

, the multiplication contributes

2

. Subtracting that from

18

gives

16

.

Worked example

Zero exponent and a negative base

Evaluate

(-3)^2+5^0-(-2)^3

.

Evaluate each power
The parentheses make $-3$ and $-2$ the bases. Squaring $-3$ gives a positive result because two negative factors multiply to a positive. The zero power of the non-zero base $5$ is $1$ . Three factors of $-2$ give a negative result.
$(-3)^2=9, 5^0=1, (-2)^3=-8$
Substitute the values
Replace each power with its value. The expression now has subtraction of a negative number, which increases the value by $8$ .
$9+1-(-8)$
Add and subtract
First add $9$ and $1$ . Then subtracting $-8$ is the same as adding $8$ , giving a total of $18$ .
$9+1+8=18$

Answer:

18

Check: Directly,

(-3)^2=9

,

5^0=1

, and

(-2)^3=-8

. Thus

9+1-(-8)=18

.

Common mistakes and how to avoid them

Treating

4^3

as

4\times3

.

Correction: The exponent counts factors of the base, so

4^3=4\times4\times4

.

Assuming a negative exponent makes the answer negative.

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

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

, which is positive.

Ignoring parentheses around a negative base.

Correction: In

(-2)^2

, the base is

-2

and the result is

4

. Parentheses determine whether the negative sign is part of the base.

Evaluating addition before a power or multiplication.

Correction: Evaluate powers first, then multiplication and division, and finally addition and subtraction.

Lesson summary

A positive exponent represents repeated multiplication of the base.
A non-zero number raised to zero equals $1$ .
A negative exponent means take the reciprocal of the corresponding positive power.
Parentheses matter when a negative number is the base.
Use the order of operations to evaluate the full expression.

Check your understanding

Question 1

What is the value of

3^{-2}

?

$-9$
$\frac{1}{9}$
$9$
$-\frac{1}{9}$

Show answer and explanation

\frac{1}{9}

A negative exponent means take the reciprocal:

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

.

Question 2

Evaluate

(-4)^2-2^0

.

$15$
$17$
$-15$
$7$

Show answer and explanation

15

The powers are

(-4)^2=16

and

2^0=1

. Their difference is

16-1=15

.

Question 3

Evaluate

10+2^3\div4

.

$3$
$12$
$16$
$18$

Show answer and explanation

12

Evaluate the power first:

2^3=8

. Then divide to get

8\div4=2

, and add:

10+2=12

.

Key terms

Base: The number that is raised to a power.
Exponent: The number that indicates how a base is used in a power.
Power: An expression made from a base and an exponent.
Reciprocal: The number that multiplies a given non-zero number to make $1$ .
Integer: A whole number, its negative, or zero.

Continue through MBF3C

View the complete MBF3C Ontario Grade 11 Mathematics curriculum and lessons

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MBF3C), expectation A2.2. 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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