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Evaluate and Use Negative Indices

In this worksheet, students will evaluate and calculate with negative indices.

'Evaluate and Use Negative Indices' worksheet

Key stage:  KS 4

Curriculum topic:  

Curriculum subtopic:  

Difficulty level:  

Worksheet Overview

QUESTION 1 of 10

In previous activities, we have worked with positive integers, and we are now going to learn to evaluate numbers using negative indices.

Think about the patterns you can see in the following table. 

 

Start with 1000 1000 10 x 10 x 10 103
 Then divide by 10 100 10 x 10 102
Divide by 10 again 10 10 101
Divide by 10 again 1 1 100  Remember that any term to the power of 0 is 1
Divide by 10 again  10-1
Divide by 10 again 10-2
Divide by 10 again 10-3

 

 

We can see that the pattern in these powers of 10 shows us that

 

Notice that we are finding the "reciprocal" of

  

For example

Evaluate 2-3 means work out the value of

 

We have found the reciprocal of 23

Evaluate 3-4 means work out the value of

We have found the reciprocal of 34

We will work out some questions like this, and then try to evaluate negative indices with negative numbers and fractions

Evaluate 2-4

( 2 + -4)

(2x2x2x2)


2 x (-4)

Evaluate 5-3

We should also be able to work out negative indices with negative numbers.

For example to work out the value of (-5)-3  we need to calculate 

 

This can also be written as 

Can you now evaluate (-6)-3

(-6 + -3)

-6 x -6 x-6


-6 x -3

We will also need to be able to evaluate powers of fractions.

For example can be calculated by finding 

So 

Here's another example 

Here is a question for you

Evaluate 

Is the correct answer

A) 

B)

C)

D)

 

A)

B)

C)

D)

We can be asked to evaluate negative fractions 

Remember that when we cube a negative number we should expect a negative answer.

 

So 

Evaluate 

8 / 27

-8 / 27

Now we need to look at using negative indices with fractions.

Remember that a negative power involves finding a reciprocal. This means that 

To work out a 1 divided by a fraction we need to change the operator to multiplication and invert or "flip" the fraction.

This means 

So the correct answer is 3

 

Here is another example

If we take this step by step 

  • we know that a negative power means that we need to find the reciprocal of the fraction to the power of 2
  • so we are calculating 1 divided by one third squared
  • we follow the rules of dividing fractions and find that 9 is the correct answer

 

Your question - Evaluate 

Hint - this means we need to work out 

So far we have found negative powers of fractions that have a numerator of 1.

We follow the same steps to calculate the negative power of a fractions that has another number as the numerator.

Follow this example - 

Evaluate 

This means find the value of 

As before, we use our fractions skills to work this out 

Informally, we can see that finding the reciprocal of the fractions has inverted the original fraction ( or turned it upside down!)

So we have found out that 

We can use the  "shortcut" when we need to work out fractions with other negative powers. 

This example shows us how to do this.

Evaluate 

As a first step we can write this as 

and we calculate 

Here's your question - Evaluate 

Hint- write this as 

Which of the following are equivalent to 

A)  

B)

C) 

D) 

A

B

C

D

Which of the following are equivalent to 

A) 

B)

C)

D)

A

B

C

D

Can you evaluate 

 

 

The answer is 

 

Do you think that this is...?

A- True

B- False

  • Question 1

Evaluate 2-4

CORRECT ANSWER
EDDIE SAYS
We have been asked to work out 2 to the power of negative 4. This is the same as 1 / (2 to the power of 4). So, 2 to the power of 4 is 16 and the reciprocal of 16 is 1/16
  • Question 2

Evaluate 5-3

CORRECT ANSWER
EDDIE SAYS
We have been asked to work out 5 to the power of negative 3. This is 1/ 5 to the power of 3. 5 to the power of 3, or 5 cubed is 125 This means the answer is
  • Question 3

We should also be able to work out negative indices with negative numbers.

For example to work out the value of (-5)-3  we need to calculate 

 

This can also be written as 

Can you now evaluate (-6)-3

CORRECT ANSWER
EDDIE SAYS
We should calculate the reciprocal of -6 to the power of 3. -6 to the power of 3 is -216, so the correct answer is
  • Question 4

We will also need to be able to evaluate powers of fractions.

For example can be calculated by finding 

So 

Here's another example 

Here is a question for you

Evaluate 

Is the correct answer

A) 

B)

C)

D)

 

CORRECT ANSWER
B)
EDDIE SAYS
We need to square 2 and square 5 and write these answers as a fraction. So
  • Question 5

We can be asked to evaluate negative fractions 

Remember that when we cube a negative number we should expect a negative answer.

 

So 

Evaluate 

CORRECT ANSWER
-8 / 27
EDDIE SAYS
Cubing a negative number gives a negative answer, so we know the correct value is
  • Question 6

Now we need to look at using negative indices with fractions.

Remember that a negative power involves finding a reciprocal. This means that 

To work out a 1 divided by a fraction we need to change the operator to multiplication and invert or "flip" the fraction.

This means 

So the correct answer is 3

 

Here is another example

If we take this step by step 

  • we know that a negative power means that we need to find the reciprocal of the fraction to the power of 2
  • so we are calculating 1 divided by one third squared
  • we follow the rules of dividing fractions and find that 9 is the correct answer

 

Your question - Evaluate 

Hint - this means we need to work out 

CORRECT ANSWER
27
EDDIE SAYS
We calculate the answer like this -
  • Question 7

So far we have found negative powers of fractions that have a numerator of 1.

We follow the same steps to calculate the negative power of a fractions that has another number as the numerator.

Follow this example - 

Evaluate 

This means find the value of 

As before, we use our fractions skills to work this out 

Informally, we can see that finding the reciprocal of the fractions has inverted the original fraction ( or turned it upside down!)

So we have found out that 

We can use the  "shortcut" when we need to work out fractions with other negative powers. 

This example shows us how to do this.

Evaluate 

As a first step we can write this as 

and we calculate 

Here's your question - Evaluate 

Hint- write this as 

CORRECT ANSWER
25 /4
EDDIE SAYS
The correct solution is
  • Question 8

Which of the following are equivalent to 

A)  

B)

C) 

D) 

CORRECT ANSWER
B
C
EDDIE SAYS
Our calculations should look like this
  • Question 9

Which of the following are equivalent to 

A) 

B)

C)

D)

CORRECT ANSWER
A
C
EDDIE SAYS
Our calculations should look like this
  • Question 10

Can you evaluate 

 

 

The answer is 

 

Do you think that this is...?

CORRECT ANSWER
B- False
EDDIE SAYS
This answer is false. Want to check? Here are the steps in calculating this question.
---- OR ----

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