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Index notation

Write expressions in index form and expanded form, and evaluate powers.

Worked examples

Writing in index form

Straightforward

Problem

Write

3 \times 3 \times 3 \times 3 \times 3

in index form.

1

Identify the base — the number being multiplied.

The number being multiplied is

3

, so the base is

3

.

2

Count how many times the base is multiplied.

3

appears

5

times in the product.

3

Write the base with the count as the exponent (index).

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

Answer

3^5

Evaluating a power

Moderate

Problem

Calculate

4^3

.

1

Write the power in expanded form.

4^3 = 4 \times 4 \times 4

2

Multiply the first two factors.

4 \times 4 = 16

3

Multiply the result by the remaining factor.

16 \times 4 = 64

Answer

64

Comparing expressions with indices

Challenging

Problem

Which is greater:

2^6

or

5^3

?

1

Evaluate

2^6

by expanding and multiplying.

2^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64

2

Evaluate

5^3

by expanding and multiplying.

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

3

Compare the two values.

125 > 64

So

5^3 > 2^6

.

Answer

5^3

is greater (

5^3 = 125

,

2^6 = 64

)

Practise

Q1·Straightforward

Write

4 \times 4 \times 4

in index form

4^n

. What is the value of

n

?

Explanation

4

is multiplied

3

times, so:

4 \times 4 \times 4 = 4^3

The value of

n

is

3

.

Q2·Straightforward

Write

2 \times 2 \times 2 \times 2 \times 2

in index form

2^n

. What is the value of

n

?

Explanation

2

is multiplied

5

times, so:

2 \times 2 \times 2 \times 2 \times 2 = 2^5

The value of

n

is

5

.

Q3·Straightforward

The expression

6^4

written in expanded form is

6 \times 6 \times 6 \times 6

. How many factors of

6

are there?

Explanation

In

6^4

, the exponent is

4

. This means

6

is multiplied

4

times:

6^4 = 6 \times 6 \times 6 \times 6

There are

4

factors of

6

.

Q4·Straightforward

Which expression represents

7 \times 7 \times 7 \times 7

in index form?

$4^7$$7^4$$7 \times 4$$28$

Explanation

The base is

7

and it is multiplied

4

times, so the index form is

7^4

.
Note:

4^7

has the base and exponent swapped, which gives a different value.

Q5·Moderate

Calculate

3^4

.

Explanation

3^4 = 3 \times 3 \times 3 \times 3

3 \times 3 = 9

9 \times 3 = 27

27 \times 3 = 81

Q6·Moderate

Calculate

2^6

.

Explanation

2^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2

= 4 \times 2 \times 2 \times 2 \times 2

= 8 \times 2 \times 2 \times 2

= 16 \times 2 \times 2

= 32 \times 2

= 64

Q7·Moderate

Calculate

5^3

.

Explanation

5^3 = 5 \times 5 \times 5

5 \times 5 = 25

25 \times 5 = 125

Q8·Moderate

Calculate

10^4

.

Explanation

10^4 = 10 \times 10 \times 10 \times 10

= 100 \times 10 \times 10

= 1000 \times 10

= 10\,000

Q9·Challenging

Which has the greatest value:

2^5

,

3^3

, or

4^2

?

$2^5$$3^3$$4^2$They are all equal.

Explanation

Evaluate each expression:

2^5 = 32

3^3 = 27

4^2 = 16

The greatest value is

32

, so

2^5

is largest.

Q10·Challenging

Evaluate

2^4

,

5^2

, and

3^3

. What is the middle value when they are ordered from smallest to largest?

Explanation

Evaluate each expression:

2^4 = 16

5^2 = 25

3^3 = 27

Ordered from smallest to largest:

16, 25, 27

.
The middle value is

25

.

Q11·Challenging

Calculate

3^4 - 2^5

.

Explanation

Evaluate each power:

3^4 = 81

2^5 = 32

Subtract:

3^4 - 2^5 = 81 - 32 = 49

Q12·Challenging

Which expression has the smallest value:

10^2

,

3^4

, or

5^3

?

$10^2$$3^4$$5^3$They are all equal.

Explanation

Evaluate each expression:

10^2 = 100

3^4 = 81

5^3 = 125

The smallest value is

81

, so

3^4

is the smallest.

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