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Zero and negative indices

Evaluate expressions with zero and negative indices, and rewrite negative index expressions using positive indices.

Worked examples

Evaluating a zero index

Straightforward

Problem

Evaluate

(-8)^0

.

1

Recall the zero index rule.

For any non-zero value

a

:

a^0 = 1

2

Apply the rule to

(-8)^0

.

(-8)^0 = 1

Answer

1

Evaluating a negative index

Moderate

Problem

Evaluate

5^{-3}

. Write your answer as a fraction.

1

Recall the negative index rule.

a^{-n} = \dfrac{1}{a^n}

2

Rewrite

5^{-3}

using a positive index.

5^{-3} = \dfrac{1}{5^3}

3

Calculate the denominator.

5^3 = 5 \times 5 \times 5 = 125

, so

5^{-3} = \dfrac{1}{125}

Answer

\dfrac{1}{125}

Combining index laws with negative indices

Challenging

Problem

Simplify

\dfrac{2^5 \times 2^{-3}}{2^4}

and write your answer as a fraction.

1

Apply the product rule to the numerator.

2^5 \times 2^{-3} = 2^{5+(-3)} = 2^2

2

Apply the quotient rule.

\dfrac{2^2}{2^4} = 2^{2-4} = 2^{-2}

3

Apply the negative index rule to write as a fraction.

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

Answer

\dfrac{1}{4}

Practise

Q1·Straightforward

What is

7^0

?

$0$$1$$7$$49$

Explanation

The zero index rule states that

a^0 = 1

for any non-zero value of

a

. So

7^0 = 1

.

Q2·Straightforward

What is

(3x)^0

, where

x \ne 0

?

$0$$1$$3$$3x$

Explanation

The zero index rule states that any non-zero expression raised to the power 0 equals 1. So

(3x)^0 = 1

.

Q3·Straightforward

What is

(-5)^0

?

$-5$$-1$$0$$1$

Explanation

For any non-zero value of

a

,

a^0 = 1

. Since

-5 \ne 0

, we have

(-5)^0 = 1

.

Q4·Straightforward

Evaluate

2^0 + 3^0

.

$0$$1$$2$$5$

Explanation

Applying the zero index rule:

2^0 = 1

and

3^0 = 1

. So

2^0 + 3^0 = 1 + 1 = 2

.

Q5·Moderate

Evaluate

2^{-3}

. Write your answer as a fraction.

/

Explanation

Using the negative index rule:

2^{-3} = \dfrac{1}{2^3} = \dfrac{1}{8}

.

Q6·Moderate

Evaluate

3^{-2}

. Write your answer as a fraction.

/

Explanation

Using the negative index rule:

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

.

Q7·Moderate

Evaluate

5^{-1}

. Write your answer as a fraction.

/

Explanation

Using the negative index rule:

5^{-1} = \dfrac{1}{5^1} = \dfrac{1}{5}

.

Q8·Moderate

Evaluate

4^{-2}

. Write your answer as a fraction.

/

Explanation

Using the negative index rule:

4^{-2} = \dfrac{1}{4^2} = \dfrac{1}{16}

.

Q9·Challenging

Simplify

2^3 \times 2^{-5}

and write your answer as a fraction.

/

Explanation

Applying the product rule:

2^3 \times 2^{-5} = 2^{3+(-5)} = 2^{-2}

. Then applying the negative index rule:

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

.

Q10·Challenging

Evaluate

\dfrac{3^4}{3^6}

. Write your answer as a fraction.

/

Explanation

Applying the quotient rule:

\dfrac{3^4}{3^6} = 3^{4-6} = 3^{-2}

. Then applying the negative index rule:

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

.

Q11·Challenging

Evaluate

(2^{-1})^3

. Write your answer as a fraction.

/

Explanation

Applying the power of a power rule:

(2^{-1})^3 = 2^{-1 \times 3} = 2^{-3}

. Then applying the negative index rule:

2^{-3} = \dfrac{1}{2^3} = \dfrac{1}{8}

.

Q12·Challenging

Evaluate

\dfrac{4^2 \times 4^{-3}}{4^{-2}}

.

Explanation

Apply the product rule to the numerator:

4^2 \times 4^{-3} = 4^{2+(-3)} = 4^{-1}

. Then apply the quotient rule:

\dfrac{4^{-1}}{4^{-2}} = 4^{-1-(-2)} = 4^{-1+2} = 4^1 = 4

.

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