A2.1 Explain zero and negative exponents | Ontario Grade 11 Mathematics

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.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MBF3C), expectation A2.1. It is a study resource, not an official curriculum publication.

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A2.1 · Explain zero and negative exponents

Ontario Grade 11 Mathematics

Ontario Grade 11 MBF3C study topic A2.1

What you will learn

1. Review: positive exponents make a pattern

2. Explain a zero exponent

3. Explain a negative exponent

4. Apply the rules carefully

Powers of 2: continue the pattern by dividing by 2

Example 1: A zero exponent

Example 2: A negative exponent with a fractional base

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MBF3C

About this lesson