Number · Number Operations · Lesson 8

Writing Numbers as Powers and Products of Powers

Numbers like 32, 81 and 1,000 hide repeated factors inside them. In this lesson you will break a number into equal factors, write it as a power such as $3^{4}$ , and write bigger numbers as a product of powers such as $2^{3} \times 3^{2}$ . This skill is the key to finding HCF and LCM and to simplifying fractions quickly.

NaCCA B7.1.2.3.2 ⏱ about 45 min

By the end you will be able to

✔Express a number such as 32 or 81 as a product of repeated factors and as a single power.
✔Write a product such as 16 × 27 as a product of two powers, $2^{4} \times 3^{3}$ .
✔Use a factor tree or repeated division to write a number as a product of prime factors in index form.
✔Avoid combining powers that have different bases.

Key words

Prime number: A whole number greater than 1 with exactly two factors, 1 and itself: 2, 3, 5, 7, 11, 13, …
Prime factor: A factor of a number that is also a prime number. The prime factors of 12 are 2 and 3.
Index form: A number written using powers, e.g. 32 = $2^{5}$ or 72 = $2^{3} \times 3^{2}$ .
Product of powers: Two or more powers multiplied together, such as $2^{4} \times 3^{3}$ .
Factor tree: A diagram that splits a number into factors again and again until every branch ends in a prime number.

Curriculum reference

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

Indicator: Express a given number as a product of a given number or numbers, as well as, in the form of a power or two such numbers as product of powers.

Starter

The salt cartons at Kejetia

At Kejetia market, Auntie Akua receives 16 cartons of salt, with 27 sachets in each carton. Her son Kwame notices that 16 = 2 × 2 × 2 × 2 and 27 = 3 × 3 × 3. "So 16 × 27 is just $2^{4} \times 3^{3}$ !" he says. Splitting numbers into small repeated pieces like this makes big numbers easy to handle. Let us learn how Kwame did it.

1. Breaking a number into equal factors

In the last lesson you turned repeated factors into a number: 2 × 2 × 2 × 2 × 2 = $2^{5}$ = 32. Now we go backwards: given 32, can we write it as a power? The trick is to keep dividing by the same number until you reach 1, and count how many times you divided.

1Choose the base you want (often the smallest number that divides exactly). For 32, choose 2.
2Divide again and again: 32 ÷ 2 = 16, 16 ÷ 2 = 8, 8 ÷ 2 = 4, 4 ÷ 2 = 2, 2 ÷ 2 = 1.
3Count the divisions: there are 5, so 32 = 2 × 2 × 2 × 2 × 2.
4Write it in index form: 32 = $2^{5}$ .

Numbers written as a product of equal factors and as a power
Number	Product of equal factors	Power
32	2 × 2 × 2 × 2 × 2	$2^{5}$
81	3 × 3 × 3 × 3	$3^{4}$
49	7 × 7	$7^{2}$
125	5 × 5 × 5	$5^{3}$
1,000	10 × 10 × 10	$1 0^{3}$

2. Writing a product as a product of powers

Some numbers are the product of two different repeated factors. Look at 16 × 27. Break each number separately: 16 = 2 × 2 × 2 × 2 = $2^{4}$ and 27 = 3 × 3 × 3 = $3^{3}$ . So

$16 \times 27 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 = 2^{4} \times 3^{3}$

The 2s and the 3s are counted separately: four 2s give $2^{4}$ and three 3s give $3^{3}$ . In the same way, 8 × 25 = (2 × 2 × 2) × (5 × 5) = $2^{3} \times 5^{2}$ .

3. Any number as a product of prime powers

Every whole number greater than 1 that is not prime can be broken down into prime factors. Writing those prime factors in index form gives a neat "fingerprint" of the number. A factor tree does this: split the number into two factors, then keep splitting every branch until each branch ends in a prime. Circle the primes.

▶ Watch 72 split into primes. Count the 2s and the 3s at the ends of the branches to get the index form.

The tree shows 72 = 2 × 2 × 2 × 3 × 3. There are three 2s and two 3s, so **72 = $2^{3} \times 3^{2}$ . You can also use repeated division** (the "ladder"): divide by the smallest prime that works, write the answer underneath, and continue until you reach 1.

Repeated division of 360 by primes
Prime divisor	Number	Result
2	360	180
2	180	90
2	90	45
3	45	15
3	15	5
5	5	1

Reading the divisors down the first column: 360 = 2 × 2 × 2 × 3 × 3 × 5 = $2^{3} \times 3^{2} \times 5$ . A prime that appears only once is written without an index (5 means $5^{1}$ ).

Worked examples

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

Example 1

A single power

Express 81 as a product of equal factors and as a power of 3.

Show the solution

Step 1Divide by 3 again and again: 81 ÷ 3 = 27, 27 ÷ 3 = 9, 9 ÷ 3 = 3, 3 ÷ 3 = 1.
Step 2We divided 4 times, so 81 = 3 × 3 × 3 × 3.
Step 3In index form: 81 = $3^{4}$ .

Answer: 81 = 3 × 3 × 3 × 3 = $3^{4}$

Example 2

A product of two powers

Express 16 × 27 as a product of powers.

Show the solution

Step 116 = 2 × 2 × 2 × 2 = $2^{4}$ .
Step 227 = 3 × 3 × 3 = $3^{3}$ .
Step 3So 16 × 27 = 2 × 2 × 2 × 2 × 3 × 3 × 3 = $2^{4} \times 3^{3}$ .

Answer: $2^{4} \times 3^{3}$

Example 3

Prime factors of 360 with a factor tree

Express 360 as a product of prime factors in index form.

▶ Follow each branch down to a prime, then group equal primes into powers.

Show the solution

Step 1360 = 2 × 180; 180 = 2 × 90; 90 = 2 × 45.
Step 245 = 3 × 15; 15 = 3 × 5, and 5 is prime.
Step 3So 360 = 2 × 2 × 2 × 3 × 3 × 5.
Step 4Count: three 2s, two 3s, one 5 → 360 = $2^{3} \times 3^{2} \times 5$ .
Step 5Check: 8 × 9 × 5 = 72 × 5 = 360 ✓

Answer: 360 = $2^{3} \times 3^{2} \times 5$

Example 4

One thousand in two ways

Express 1,000 (a) as a power of 10 and (b) as a product of prime factors in index form.

Show the solution

Step 1(a) 1,000 = 10 × 10 × 10 = $1 0^{3}$ .
Step 2(b) Each 10 = 2 × 5, so 1,000 = (2 × 5) × (2 × 5) × (2 × 5).
Step 3Collect the 2s and the 5s: 2 × 2 × 2 × 5 × 5 × 5 = $2^{3} \times 5^{3}$ .
Step 4Check: 8 × 125 = 1,000 ✓

Answer: (a) $1 0^{3}$ (b) $2^{3} \times 5^{3}$

Summary

To write a number as a power, keep dividing by the base and count the divisions: 32 = $2^{5}$ , 81 = $3^{4}$ , 49 = $7^{2}$ .
For a product such as 16 × 27, break each number separately: $2^{4} \times 3^{3}$ .
A factor tree or repeated division writes any number as a product of prime powers, e.g. 72 = $2^{3} \times 3^{2}$ .
Never combine powers with different bases: $2^{4} \times 3^{3}$ ≠ $6^{7}$ .
Always check by multiplying the factors back together.

The salt cartons at Kejetia

1. Breaking a number into equal factors

2. Writing a product as a product of powers

3. Any number as a product of prime powers

Worked examples

A single power

A product of two powers

Prime factors of 360 with a factor tree

One thousand in two ways

Summary

Practice: 15 questions