Basic 7 · Mathematics · Number Operations
Expressing Numbers as Products of Powers and the Zero Index
In this topic you will express numbers as products of factors and of powers, write numbers as products of prime factors in index form, and show that any non-zero number raised to the power zero equals one.
By the end of these notes you should be able to:
- Express a number as a product of factors and as a power
- Write numbers as products of prime factors in index form
- Show that a⁰ = 1 for any non-zero number a and use it to solve problems
What you already know
- Repeated factors and powers
- Factors and prime numbers (Upper Primary)
Copy everything that has a blue line on its left into your notebook. The other parts are for reading.
Lesson 1
- Strand
- Number
- Sub-strand
- Number Operations
- Indicator
- B7.1.2.3.2 – 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
Introduction
The number 64 can be written as 8 × 8, as 4 × 4 × 4, or as 2 × 2 × 2 × 2 × 2 × 2. Each way shows a different structure of the same number.
1. Numbers as Products of Factors
A factor is a number that divides another exactly. Any composite number can be written as a product of factors in different ways: 36 = 4 × 9 = 6 × 6 = 2 × 18 = 3 × 12.
2. Numbers as Powers
| Number | As a power | Other power forms |
|---|---|---|
| 64 | 8² | 4³ or 2⁶ |
| 81 | 9² | 3⁴ |
| 1,000 | 10³ | |
| 125 | 5³ | |
| 256 | 16² | 4⁴ or 2⁸ |
3. Products of Powers
Some numbers are products of two different powers: 72 = 8 × 9 = 2³ × 3². Here, 2³ = 8 and 3² = 9. Similarly, 200 = 8 × 25 = 2³ × 5².
Express as a product of powers
Express 108 as a product of powers.
- 108 = 4 × 27.
- 4 = 2² and 27 = 3³.
Answer: 108 = 2² × 3³
4. Checking Your Answer
Always multiply back to check: 2² × 3³ = 4 × 27 = 108. If the product is not the original number, there is a mistake.
5. Why Products of Powers Are Useful
Writing numbers as products of powers makes large calculations easier and shows hidden patterns. For example, knowing that 1,000 = 10³ and 1,000,000 = 10⁶ helps us see that a million is a thousand thousands. In later topics, products of powers help us find the HCF and LCM, simplify fractions and work with very large or very small numbers in science.
New words
- factor
- A number that divides another exactly.
- product
- The result of multiplication.
- composite number
- A number with more than two factors.
- product of powers
- A number written as powers multiplied together.
Summary
- Composite numbers can be written as products of factors in several ways.
- Some numbers can be written as a single power, such as 81 = 3⁴.
- Others are products of powers, such as 72 = 2³ × 3²; check by multiplying.
Class activity: Power Puzzles
- Write 16, 27, 32, 100 and 144 as powers or products of powers.
- Check each by multiplying.
- Challenge your partner with a new number.
You need: Bottle tops, Bundles of sticks, Printed factor-and-power cards
Exercises
Answer in your exercise book first, then check your answers.
1. 49 written as a power is…
- A. 7²
- B. 4⁹
- C. 9²
- D. 2⁷
Show answer
Answer: 7²7 × 7.2. 2³ × 5² =
- A. 200
- B. 100
- C. 40
- D. 250
Show answer
Answer: 2008 × 25.3. 81 = 3⁴.
- A. True
- B. False
Show answer
Answer: True3 × 3 × 3 × 3.4. A number that divides another exactly is a ______.
Show answer
Answer: factorDivisor.5. Write 64 as a power in two different ways.
Show answer
Answer: 8² and 2⁶ (or 4³).Any two.6. Express 72 as a product of powers of 2 and 3.
Show answer
Answer: 2³ × 3²8 × 9.
Lesson 2
- Strand
- Number
- Sub-strand
- Number Operations
- Indicator
- B7.1.2.3.2 – 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
Introduction
Every composite number can be broken down into prime factors, like a building made of bricks. Writing these prime factors in index form gives a short, unique description of the number.
1. Prime Numbers
Prime number: A number with exactly two factors: 1 and itself, such as 2, 3, 5, 7, 11 and 13.
2. Factor Trees
Factor tree
Express 360 as a product of prime factors.
- 360 = 36 × 10.
- 36 = 6 × 6 = 2 × 3 × 2 × 3.
- 10 = 2 × 5.
- Collect primes: 2 × 2 × 2 × 3 × 3 × 5.
Answer: 360 = 2³ × 3² × 5
3. Repeated Division
Repeated division
Express 252 as a product of prime factors.
- 252 ÷ 2 = 126.
- 126 ÷ 2 = 63.
- 63 ÷ 3 = 21.
- 21 ÷ 3 = 7.
- 7 ÷ 7 = 1.
Answer: 252 = 2² × 3² × 7
4. Uniqueness
Every composite number has only one set of prime factors, no matter how you start the factor tree. For 360, you could start with 2 × 180 or 12 × 30, but you will always end with 2³ × 3² × 5. This is why prime factorisation is so useful.
5. Divisibility Tests to Help
Divisibility tests speed up factor trees. A number is divisible by 2 if its last digit is even; by 3 if the sum of its digits is divisible by 3; by 5 if it ends in 0 or 5; and by 9 if the sum of its digits is divisible by 9. For 252, the digit sum is 2 + 5 + 2 = 9, so it is divisible by both 3 and 9.
New words
- prime number
- A number with exactly two factors.
- prime factorisation
- Writing a number as a product of primes.
- factor tree
- A diagram that breaks a number into factors.
- index form
- Using powers to show repeated factors.
Summary
- Prime numbers have exactly two factors.
- Use a factor tree or repeated division to find prime factors.
- Write repeated prime factors in index form; the result is unique for each number.
Class activity: Factor Tree Gallery
- Each group draws factor trees for three numbers.
- Write each result in index form.
- Compare trees that start differently but give the same result.
You need: Bottle tops, Bundles of sticks, Printed factor-and-power cards
Exercises
Answer in your exercise book first, then check your answers.
1. Which is a prime number?
- A. 13
- B. 15
- C. 21
- D. 27
Show answer
Answer: 13Only 1 and 13.2. 120 as a product of primes is…
- A. 2³ × 3 × 5
- B. 2² × 3 × 5
- C. 2³ × 15
- D. 2 × 3 × 4 × 5
Show answer
Answer: 2³ × 3 × 58 × 15.3. 1 is a prime number.
- A. True
- B. False
Show answer
Answer: FalseIt has only one factor.4. Writing a number as a product of primes is prime ______.
Show answer
Answer: factorisationBreaking down.5. Express 90 as a product of prime factors in index form.
Show answer
Answer: 2 × 3² × 52 × 45 = 2 × 9 × 5.6. Show that two different factor trees for 60 give the same result.
Show answer
Answer: 6 × 10 → 2 × 3 × 2 × 5; 4 × 15 → 2 × 2 × 3 × 5; both 2² × 3 × 5.Uniqueness.
Lesson 3
- Strand
- Number
- Sub-strand
- Number Operations
- Indicator
- B7.1.2.3.3 – Show that the value of any natural number with zero as its exponent or index is 1 and use it to solve problems
Introduction
What is 5⁰? It may seem strange, but any non-zero number raised to the power zero equals 1. Let us discover why.
1. Discovering the Pattern
| Power | Value |
|---|---|
| 2⁴ | 16 |
| 2³ | 8 |
| 2² | 4 |
| 2¹ | 2 |
| 2⁰ | ? |
Each time the index decreases by 1, the value is divided by 2: 16 → 8 → 4 → 2. Following the pattern, 2⁰ = 2 ÷ 2 = 1.
2. Using Division
We know that a number divided by itself equals 1: 5³ ÷ 5³ = 125 ÷ 125 = 1. When dividing powers with the same base, we subtract the indices: 5³ ÷ 5³ = 5³⁻³ = 5⁰. Since both answers must be equal, 5⁰ = 1.
3. The Rule
Zero index rule: Any non-zero number raised to the power 0 equals 1: a⁰ = 1, where a ≠ 0.
Examples: 7⁰ = 1, 100⁰ = 1, 1,000,000⁰ = 1, (½)⁰ = 1. Note that 0⁰ is not defined at this level.
4. Solving Problems
Evaluate
Find the value of 4⁰ + 3² × 2⁰.
- 4⁰ = 1.
- 3² = 9 and 2⁰ = 1, so 3² × 2⁰ = 9.
- 1 + 9 = 10.
Answer: 10
Find the unknown
If 6ⁿ = 1, find n.
- Any non-zero number to the power 0 is 1.
Answer: n = 0
5. The Index of 1
A related rule is that any number to the power 1 is the number itself: a¹ = a. So 7¹ = 7 and 250¹ = 250. Together with the zero index rule, this completes the pattern: 3² = 9, 3¹ = 3 and 3⁰ = 1. Each step down in the index divides the value by the base.
New words
- zero index
- An index (power) of 0.
- pattern
- A regular sequence that follows a rule.
- evaluate
- To find the value of.
- non-zero
- Not equal to zero.
Summary
- The pattern of powers shows that 2⁰ = 1.
- Dividing a power by itself gives a⁰ = 1.
- Any non-zero number to the power zero equals 1.
Class activity: Pattern Investigation
- Write powers of 3 from 3⁴ down to 3¹ and their values.
- Continue the pattern to find 3⁰.
- Explain your reasoning to the class.
You need: Number cards, Square-grid paper, Calculator
Exercises
Answer in your exercise book first, then check your answers.
1. 9⁰ =
- A. 1
- B. 0
- C. 9
- D. 90
Show answer
Answer: 1Zero index.2. 5⁰ + 5¹ =
- A. 6
- B. 5
- C. 1
- D. 10
Show answer
Answer: 61 + 5.3. 1,000⁰ = 1.
- A. True
- B. False
Show answer
Answer: TrueAny non-zero number.4. If aⁿ = 1 and a is not 0 or 1, then n = ______.
Show answer
Answer: 0Zero index.5. Evaluate 2³ × 7⁰.
Show answer
Answer: 88 × 1.6. Use division to show that 4⁰ = 1.
Show answer
Answer: 4² ÷ 4² = 16 ÷ 16 = 1, and 4² ÷ 4² = 4²⁻² = 4⁰, so 4⁰ = 1.Division of powers.
Homework
- Express 48, 100 and 144 as products of prime factors in index form.
- Evaluate five expressions that include the zero index.