 # Evaluate Positive Indices

In this worksheet, students will find the value of numbers when positive and negative integer indices (or powers) are used. Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   Pearson Edexcel, OCR, Eduqas, AQA

Curriculum topic:   Number, Indices and Surds

Curriculum subtopic:   Structure and Calculation, Powers and Roots

Difficulty level:   ### QUESTION 1 of 10

In this activity, we are going to learn to evaluate numbers that are given using a positive index number.

For example, 4 squared is 16 and 2 cubed is 8.

We could write these as 42 = 16 or 4 x 4 = 16 and 23 = 8 or 2 x 2 x 2 = 8.

42  - The small 2 tells us to multiply 4 BY ITSELF two times.

The is the index number.

2 -  The small 3 tells us to multiply 2 BY ITSELF three times.

The is the index number.

If we are asked to "evaluate" (find the value of) 34 we would multiply by itself times.

This means 3 x 3 x 3 x 3 = 81

Question: Evaluate 34

Let's try another question now.

Evaluate 53.

We work out multiplied by itself times.

5 x 5 x 5 = 125

We are now going to work out (-5)3.

We will need to multiply (-5) by itself 3 times and we will need to follow the rules for multiplying negative numbers

This means we need to calculate: (-5) x (-5) x (-5)

The first step is to find (-5) x (-5)

Two negative numbers multiply to make a positive number, so the answer to this step is +25.

The next step is to multiply this result by (-5).

25 x (-5) = -125

In this activity, we will evaluate positive and negative indices to work out the whole number value of each and whether the outcome is positive or negative.

You will need to have your scientific calculator handy for some of the questions in this activity.

Evaluate:

32

(2 x 2 x 2)

(3 x 3)

(3 x 2)

(3 + 2)

Evaluate:

52

(5 + 5)

(5 + 2)

(5 x 5)

(2 x 2 x 2 x 2 x 2)

Evaluate:

34

(3 x 4)

(4 x 4 x 4)

(3 + 4)

(3 x 3 x 3 x 3)

Evaluate:

54

e.g. Using your calculator, evaluate 84.

The power button on your calculator means you can ask it to work out 8 x 8 x 8 x 8 in one step.

Give this a try now.

Enter 8 as the base of the power and 4 as the index number and then press "=".

The correct answer is 4096, which you should see on your calculator.

Using this same process, evaluate:

94

13

6561

36

e.g. Evaluate (-3)2

To solve this, we need to find the value of (-3) x (-3).

We will need to remember that multiplying a negative number by another negative number gives a positive answer

This means -3 x -3 = 9 (positive 9)

Using this same process, evaluate:

(-5)2

Positive 25

Negative 25

Evaluate:

(-4)3

Negative 64

Positive 64

(-5)6

Remember to use brackets to tell your calculator to multiply negative 5 by itself 6 times.

Typing in just -56 without brackets will ask your calculator to find 56 and then make the answer negative.

Positive 15625

Negative 15625

You will need to just remember that any number raised to the power of 0 has the answer 1.

This is an important fact to commit to your memory when working with powers.

e.g. 60 = 1 and 1210 = 1

Using this knowledge, evaluate:

70

Test all the skills you have learnt in this activity by evaluating (without a calculator):

32 + (-2)3 + 40

• Question 1

Evaluate:

32

(3 x 3)
EDDIE SAYS
'Evaluate 3²' means that we need to calculate the value of 3 x 3. The index number 2 tells us to multiply 3 by itself two times. 3 x 3 = 9
• Question 2

Evaluate:

52

(5 x 5)
EDDIE SAYS
'Evaluate 5²' means that we need to work out the value of 5 x 5. The index number 2 tells us to multiply 5 by itself two times. 5 x 5 = 25
• Question 3

Evaluate:

34

(3 x 3 x 3 x 3)
EDDIE SAYS
'Evaluate 34' means find the value of: 3 x 3 x 3 x 3 The index number 4 tells us to multiply 3 by itself four times. 3 x 3 x 3 x 3 = 81
• Question 4

Evaluate:

54

625
EDDIE SAYS
To find the value of 54, we need to work out: 5 x 5 x 5 x 5 The answer to this is 625. Evaluating index numbers can also be worked out on a scientific calculator using the power button. Why not give this a try now if you have not tried this before?
• Question 5

e.g. Using your calculator, evaluate 84.

The power button on your calculator means you can ask it to work out 8 x 8 x 8 x 8 in one step.

Give this a try now.

Enter 8 as the base of the power and 4 as the index number and then press "=".

The correct answer is 4096, which you should see on your calculator.

Using this same process, evaluate:

94

6561
EDDIE SAYS
The calculator will work out (9 x 9 x 9 x 9) for you in one step and show you an answer of 6561. We will now move on to finding powers of negative numbers.
• Question 6

e.g. Evaluate (-3)2

To solve this, we need to find the value of (-3) x (-3).

We will need to remember that multiplying a negative number by another negative number gives a positive answer

This means -3 x -3 = 9 (positive 9)

Using this same process, evaluate:

(-5)2

Positive 25
EDDIE SAYS
When we calculate (-5) x (-5), we are multiplying two negative numbers together. Multiplying two negatives together, gives a positive answer. Therefore: (-5) x (-5) = 25 (or positive 25)
• Question 7

Evaluate:

(-4)3

Negative 64
EDDIE SAYS
When we multiply negative 4 by itself 3 times, we need to work out: (-4) x (-4) x (-4) As a first step, we need to multiply the first 2 numbers only: (-4) x (-4) = 16 Remember that multiplying two negatives, gives a positive answer overall. This result then needs multiplying by (-4) again. 16 x (-4) = (-64) Remember that a positive number multiplied by a negative number gives a negative result. As a top tip when working with negative integers, if the power or index is an even number then the outcome will be positive overall, but if it is odd then the outcome will be negative.
• Question 8

(-5)6

Remember to use brackets to tell your calculator to multiply negative 5 by itself 6 times.

Typing in just -56 without brackets will ask your calculator to find 56 and then make the answer negative.

Positive 15625
EDDIE SAYS
Our final answer is positive as we are multiplying a negative number by itself an even number of times. If we work this through to check: (-5) x (-5) = 25 25 x (-5) = -125 -125 x (-5) = 625 625 x (-5) = -3125 -3125 x (-5) = 15625 That confirms it then!
• Question 9

You will need to just remember that any number raised to the power of 0 has the answer 1.

This is an important fact to commit to your memory when working with powers.

e.g. 60 = 1 and 1210 = 1

Using this knowledge, evaluate:

70

1
EDDIE SAYS
Remember that any number raised to the power of 0 is 1.
• Question 10

Test all the skills you have learnt in this activity by evaluating (without a calculator):

32 + (-2)3 + 40

2
EDDIE SAYS
This may look complex, but we just need to work through each index, one step at a time. 3² = 3 x 3 = 9 (-2)³ = (-2) x (-2) x (-2) = -8 40 = 1 9 + (-8) + 1 = 2 Well done for working hard on index numbers, hopefully you are feeling much more confident with these now. Why not try an activity focused only on negative indices now?
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