Number · Number Operations · Lesson 10

Finding the Value of a Number in Index Form

A power such as $5^{3}$ is a short way of writing a number, but what number is it? In this lesson you will expand powers into repeated factors, multiply them step by step (with smart shortcuts), and evaluate expressions such as $2^{3} \times 3^{2}$ and $2^{4} + 3^{3} - 5^{0}$ . You will also use powers to find areas of square plots and volumes of cube-shaped tanks.

NaCCA B7.1.2.3.4 ⏱ about 45 min

By the end you will be able to

✔Find the value of a power by expanding it, e.g. $5^{3}$ = 5 × 5 × 5 = 125.
✔Use pairing to multiply quickly, e.g. $3^{4}$ = 9 × 9 = 81.
✔Evaluate expressions containing several powers, doing powers before ×, ÷, + and −.
✔Solve real-life problems involving squares and cubes.

Key words

Value of a power: The ordinary number a power stands for. The value of $2^{5}$ is 32.
Evaluate: To find the value of. "Evaluate $4^{3}$ " means work out 4 × 4 × 4 = 64.
Square number: A number of the form $n^{2}$ , such as 1, 4, 9, 16, 25, …
Cube number: A number of the form $n^{3}$ , such as 1, 8, 27, 64, 125, …

Curriculum reference

Content standard: B7.1.2.3 Demonstrate understanding and the use of powers of natural numbers in solving problems.

Indicator: Find the value of a number written in index form.

Starter

How big is a cube-shaped tank?

The headmaster of a JHS in Wa wants a cube-shaped concrete water tank, 3 metres long, 3 metres wide and 3 metres deep. The contractor says it will hold $3^{3}$ cubic metres of water. Is that 9? 27? 33? And how many litres is it? Knowing how to find the value of a power answers questions like this in seconds.

1. Expand, then multiply step by step

To find the value of a power, first expand it: write the base as a factor as many times as the index says. Then multiply two numbers at a time, from left to right, keeping a running total.

1Expand: $5^{3}$ = 5 × 5 × 5.
2Multiply the first two factors: 5 × 5 = 25.
3Multiply the result by the next factor: 25 × 5 = 125.
4So the value of $5^{3}$ is 125.

Pairing shortcut. When the index is even, group the factors into equal pairs. $3^{4}$ = (3 × 3) × (3 × 3) = 9 × 9 = 81. Similarly $2^{6}$ = (2 × 2 × 2) × (2 × 2 × 2) = 8 × 8 = 64. Fewer steps means fewer chances to slip.

2. Squares, cubes and powers of 2 worth knowing

Good mathematicians remember the most common powers, the same way you remember times tables. Learn this table; it will save you time in the BECE.

Squares, cubes and powers of 2
n	$n^{2}$	$n^{3}$	$2^{n}$
1	1	1	2
2	4	8	4
3	9	27	8
4	16	64	16
5	25	125	32
6	36	216	64
7	49	343	128
8	64	512	256
9	81	729	512
10	100	1000	1,024

Notice how fast each column grows. Squares grow steadily, cubes grow faster, and powers of 2 start slowly but soon overtake everything: $2^{10}$ = 1,024, more than $1 0^{3}$ . Also notice that $n^{2}$ is the area of a square with side $n$ , and $n^{3}$ is the volume of a cube with edge $n$ . That is why we say "squared" and "cubed".

3. Bigger bases: use column multiplication

For a base such as 12, the numbers grow quickly, so write the working in columns. $1 2^{3}$ = 12 × 12 × 12. First 12 × 12 = 144. Then multiply 144 by 12 in columns, as the animation shows.

▶ Watch 144 × 12 worked in columns. This last step gives the value of $1 2^{3}$ .

So $1 2^{3}$ = 1,728. Always keep the intermediate results (here 144) written down so you can check them.

4. Expressions with more than one power

In an expression, work out the powers first, then multiplication and division, then addition and subtraction (BODMAS: the "O" is "of/orders", which includes powers).

$2^{3} \times 3^{2}$ = 8 × 9 = 72
$2^{4} + 3^{3} - 5^{0}$ = 16 + 27 − 1 = 42
$1 0^{3} \div 5^{2}$ = 1,000 ÷ 25 = 40

When the bases are the same, you can use the laws of indices first and then evaluate once: $2^{3} \times 2^{2} = 2^{3 + 2} = 2^{5}$ = 32, and $1 0^{5} \div 1 0^{2} = 1 0^{5 - 2} = 1 0^{3}$ = 1,000.

Worked examples

Try each one yourself first, then open the solution and compare every step.

Example 1

Value of a cube

Find the value of

5^{3}

Show the solution

Step 1Expand: $5^{3}$ = 5 × 5 × 5.
Step 25 × 5 = 25.
Step 325 × 5 = 125.

Answer: 125

Example 2

Using pairs

Find the value of

3^{4}

Show the solution

Step 1Expand: $3^{4}$ = 3 × 3 × 3 × 3.
Step 2Pair the factors: (3 × 3) × (3 × 3) = 9 × 9.
Step 39 × 9 = 81.

Answer: 81

Example 3

A larger cube

Find the value of

6^{3}

▶ See the carrying in 36 × 6, the final step of $6^{3}$ .

Show the solution

Step 1Expand: $6^{3}$ = 6 × 6 × 6.
Step 26 × 6 = 36.
Step 336 × 6: 6 × 6 = 36 (write 6, carry 3); 6 × 3 = 18, plus 3 = 21.
Step 4So 36 × 6 = 216.

Answer: 216

Example 4

An expression with powers

Evaluate

2^{4} + 3^{3} - 5^{0}

Show the solution

Step 1Powers first: $2^{4}$ = 2 × 2 × 2 × 2 = 16.
Step 2 $3^{3}$ = 3 × 3 × 3 = 27.
Step 3 $5^{0}$ = 1 (zero index).
Step 4Now 16 + 27 − 1 = 43 − 1 = 42.

Answer: 42

Example 5

The school water tank

A cube-shaped tank is 3 m long, 3 m wide and 3 m deep. Find its volume in m³ and in litres (1 m³ = 1,000 litres).

Show the solution

Step 1Volume of a cube = side × side × side = $3^{3}$ .
Step 2 $3^{3}$ = 3 × 3 × 3 = 9 × 3 = 27 m³.
Step 3In litres: 27 × 1,000 = 27,000 litres.

Answer: 27 m³ = 27,000 litres

Summary

To evaluate a power, expand it and multiply two numbers at a time: $5^{3}$ = 25 × 5 = 125.
Pair factors to save steps: $3^{4}$ = 9 × 9 = 81; $2^{6}$ = 8 × 8 = 64.
The index is NOT a multiplier: $5^{3}$ ≠ 15.
In expressions, do powers first, then × and ÷, then + and −.
With the same base, use $a^{m} \times a^{n} = a^{m + n}$ and $a^{m} \div a^{n} = a^{m - n}$ before evaluating.

How big is a cube-shaped tank?

1. Expand, then multiply step by step

2. Squares, cubes and powers of 2 worth knowing

3. Bigger bases: use column multiplication

4. Expressions with more than one power

Worked examples

Value of a cube

Using pairs

A larger cube

An expression with powers

The school water tank

Summary

Practice: 15 questions