Explain zero and negative exponents

Ontario Grade 11 Mathematics

Mathematical Models

Ontario Grade 11 MBF3C study topic A2.1

An exponent tells how many times a base is used as a factor. For example, in

4^3

, the base is

4

and the exponent is

3

: multiply three factors of

4

. This lesson focuses on what happens when the exponent is zero or negative. The key idea is to look at the pattern made by powers of the same non-zero base.

What you will learn

Explain what a zero exponent means for a non-zero base.
Explain what a negative exponent means using repeated division and reciprocals.
Evaluate numerical expressions with zero and negative exponents.

1. Review: positive exponents make a pattern

A power is an expression with a base and an exponent. In

5^3

, the base is

5

and the exponent is

3

. A positive exponent tells how many equal factors to multiply.

To see what zero and negative exponents mean, list powers of

2

and move down one exponent at a time. Each time the exponent decreases by

1

, divide the previous value by

2

. The pattern continues through exponent zero and into negative exponents.

This pattern gives meaning to the new exponents. They are not extra multiplication instructions: a negative exponent signals a reciprocal, which means the value is flipped to make a fraction.

2^3=8,\quad 2^2=4,\quad 2^1=2

A positive exponent counts equal factors in a product.
When the exponent drops by $1$ , divide the power by the base.
Continue the same pattern through zero and negative exponents.

2. Explain a zero exponent

Continue the pattern by dividing

2^1

by

2

. The result is

1

, so

2^0=1

. The same pattern works for any non-zero base: a power with exponent zero equals

1

.

The base must not be zero. The pattern would require dividing by the base as the exponent decreases. Division by zero is not defined, so

0^0

is not covered by this rule.

A useful way to explain the rule is to compare two neighbouring powers. For a non-zero base, the value at exponent zero is the value at exponent one divided by the base. That gives

a^0=a/a=1

when

a\ne0

.

a^0=1\quad(a\ne0)

For every non-zero base, a zero exponent gives $1$ .
The zero-exponent rule does not apply to a base of zero.
Explain the rule by dividing the exponent-one power by its base.

3. Explain a negative exponent

After

2^0=1

, divide by

2

again to reach the next power:

2^{-1}=1/2

. Divide by

2

once more to get

2^{-2}=1/4

. So a negative exponent means take the reciprocal of the corresponding positive power.

A reciprocal is a number that gives

1

when multiplied by the original non-zero number. For example, the reciprocal of

2^3=8

is

1/8

, so

2^{-3}=1/8

. The exponent does not make the answer negative; the minus sign tells you to use a reciprocal.

For a non-zero base

a

and a positive whole number

n

,

a^{-n}=1/a^n

. First find the positive power, then place it in the denominator of

1

. Keep the base and exponent together when identifying the power.

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

A negative exponent means use a reciprocal, not a negative answer.
Find the matching positive power, then write $1$ divided by that power.
The base cannot be zero because a reciprocal with zero in the denominator is undefined.

4. Apply the rules carefully

When evaluating an expression with a zero or negative exponent, identify the base first. Then apply the matching rule. If the base is a fraction, the reciprocal reverses the numerator and denominator.

Check whether the result makes sense by comparing it with the pattern. For a base greater than

1

, positive powers grow as the exponent increases, while negative powers are fractions between

0

and

1

. For a base that is a positive fraction, the pattern moves in the opposite size direction.

These checks help catch common errors. The negative exponent does not put a minus sign in front of the answer, and a zero exponent does not make the answer zero.

Identify the full base before applying an exponent rule.
A negative exponent changes a power to its reciprocal.
Use the pattern of powers to check whether the answer is reasonable.

Powers of 2: continue the pattern by dividing by 2

Exponent	Power	Value
3	$2^3$	$8$
2	$2^2$	$4$
1	$2^1$	$2$
0	$2^0$	$1$
$-1$	$2^{-1}$	$1/2$
$-2$	$2^{-2}$	$1/4$

Worked example

Example 1: A zero exponent

Evaluate

9^0

and explain why.

Identify the base
The base is $9$ , which is non-zero, so the zero-exponent rule applies.
$9$
Use the pattern
The power with exponent zero is the power with exponent one divided by the base. Since $9^1=9$ , divide $9$ by $9$ .
$9^0=\frac{9^1}{9}=\frac{9}{9}=1$

Answer:

9^0=1

.

Check: The result follows the pattern: reducing the exponent from

1

to

0

divides

9

by its base,

9

.

Worked example

Example 2: A negative exponent with a fractional base

Evaluate

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

.

Take the reciprocal
A negative exponent means take the reciprocal of the base, then use the matching positive exponent. The reciprocal of $2/5$ is $5/2$ .
$(\frac{2}{5})^{-2}=(\frac{5}{2})^2$
Evaluate the positive power
Square the numerator and denominator because the exponent is $2$ .
$(\frac{5}{2})^2=\frac{5^2}{2^2}=\frac{25}{4}$

Answer:

\left(\frac{2}{5}\right)^{-2}=\frac{25}{4}

.

Check: The original base is less than

1

. Moving to a negative exponent makes the value greater than

1

, which agrees with

25/4

.

Common mistakes and how to avoid them

Treating

a^0

as

0

.

Correction: For a non-zero base,

a^0=1

. The pattern reaches

1

when the exponent decreases from one to zero.

Treating a negative exponent as a negative value, such as

2^{-3}=-8

.

Correction: A negative exponent means reciprocal:

2^{-3}=1/2^3=1/8

.

Applying the zero-exponent or negative-exponent rule to a base of zero.

Correction: These rules require a non-zero base. Division by zero is not defined.

Taking the reciprocal of only the exponent or changing the sign of the base.

Correction: Take the reciprocal of the entire non-zero base, then use the positive exponent.

Lesson summary

For a non-zero base, exponent zero gives $1$ .
A negative exponent means take the reciprocal of the corresponding positive power.
For $a\ne0$ , $a^{-n}=1/a^n$ .
A negative exponent does not automatically make a value negative.

Check your understanding

Question 1

What is

12^0

?

$0$
$1$
$12$
$-1$

Show answer and explanation

1

The base is non-zero, so any zero exponent gives

1

.

Question 2

What is

3^{-2}

?

$-9$
$9$
$1/9$
$-1/9$

Show answer and explanation

1/9

Use the reciprocal of the matching positive power:

3^{-2}=1/3^2=1/9

.

Question 3

What is

\left(\frac{1}{4}\right)^{-1}

?

$-4$
$1/4$
$4$
$0$

Show answer and explanation

4

Exponent

-1

means take the reciprocal of the base. The reciprocal of

1/4

is

4

.

Key terms

Base: The number that is raised to an exponent.
Exponent: The number that indicates a power of a base.
Power: An expression made from a base and an exponent, or the value of that expression.
Reciprocal: For a non-zero number, the value that gives $1$ when multiplied by that number.

Continue through MBF3C

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MBF3C), expectation A2.1. 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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