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Basic 7 · Mathematics · Number Operations

Evaluating Powers and Finding the HCF

In this topic you will find the value of numbers written in index form, evaluate expressions involving powers, and use prime factorisation to find the highest common factor of numbers.

By the end of these notes you should be able to:

  • Find the value of numbers written in index form
  • Evaluate expressions with powers using the correct order of operations
  • Find the HCF of numbers using prime factors in index form

What you already know

  • Powers, prime factors and the zero index

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
B7.1.2.3.4 – Find the value of a number written in index form

Introduction

Index form is short, but sometimes we need the full value. To find it, we multiply the base by itself as many times as the index says.

1. Evaluating Powers

Values of some powers
PowerWorkingValue
3⁴3 × 3 × 3 × 381
2⁷2 × 2 × 2 × 2 × 2 × 2 × 2128
5³5 × 5 × 5125
10⁶10 × 10 × 10 × 10 × 10 × 101,000,000
1⁹1 × 1 × ... × 11

2. Powers of Fractions and Decimals

Bases can be fractions or decimals: (½)³ = ½ × ½ × ½ = ⅛ and 0.2² = 0.2 × 0.2 = 0.04. Notice that when the base is between 0 and 1, the value gets smaller as the index increases.

3. Using a Calculator

Many calculators have a power key, marked xʸ, yˣ or ^. To find 3⁴, press 3, then the power key, then 4, then =. The display shows 81. Use the calculator to check answers, but practise working out small powers by hand.

A close-up of a scientific calculator keypad
Figure 1: Calculator power key Use a calculator to find 2¹⁰ and 3⁶.

4. Comparing Powers

Compare

Which is greater: 2⁵ or 5²?

  1. 2⁵ = 32.
  2. 5² = 25.

Answer: 2⁵ > 5²

5. Powers of Ten and Large Numbers

Powers of ten are evaluated simply by writing 1 followed by as many zeros as the index: 10⁴ = 10,000 and 10⁹ = 1,000,000,000. Scientists use them to write very large numbers in a short form. For example, the distance from the Earth to the Sun is about 150,000,000 km, which can be written as 1.5 × 10⁸ km.

New words

evaluate
To find the value of.
index form
A number written as a base and an index.
power key
A calculator key for raising a number to a power.
base
The number being multiplied repeatedly.

Summary

  • To evaluate a power, multiply the base by itself as many times as the index shows.
  • Powers of fractions and decimals between 0 and 1 get smaller as the index increases.
  • A calculator's power key can find and check powers.

Class activity: Power Ladder

  1. Write powers of 2 from 2⁰ to 2¹⁰ and their values.
  2. Check with a calculator.
  3. Find the first power of 2 greater than 1,000.

You need: Number cards, Square-grid paper, Calculator

Exercises

Answer in your exercise book first, then check your answers.

  1. 1. The value of 6² is…

    • A. 36
    • B. 12
    • C. 8
    • D. 66
    Show answer
    Answer: 366 × 6.

  2. 2. (⅓)² =

    • A. ⅑
    • B. ⅔
    • C. ⅙
    • D. 9
    Show answer
    Answer: ⅑⅓ × ⅓.

  3. 3. 2⁵ is greater than 5².

    • A. True
    • B. False
    Show answer
    Answer: True32 > 25.

  4. 4. To find the value of a power is to ______ it.

    Show answer
    Answer: evaluateFind its value.

  5. 5. Evaluate 4³ and 0.3².

    Show answer
    Answer: 64 and 0.09Powers.

  6. 6. Which is greater, 3⁴ or 4³? Show working.

    Show answer
    Answer: 3⁴ = 81 and 4³ = 64, so 3⁴ is greater.Compare values.

Lesson 2

Strand
Number
Sub-strand
Number Operations
Indicator
B7.1.2.3.4 – Find the value of a number written in index form

Introduction

Expressions such as 2³ + 4 × 3² mix powers with other operations. To get the correct answer, we must follow the order of operations.

1. Order of Operations (BODMAS)

BODMAS
LetterMeaningOrder
Bbracketsfirst
Oorders (powers and roots)second
D, Mdivision and multiplicationthird, from left to right
A, Saddition and subtractionlast, from left to right

2. Worked Examples

Evaluate

Find the value of 2³ + 4 × 3².

  1. Orders: 2³ = 8 and 3² = 9.
  2. Multiplication: 4 × 9 = 36.
  3. Addition: 8 + 36 = 44.

Answer: 44

Evaluate with brackets

Find the value of (5 − 2)³ ÷ 9.

  1. Brackets: 5 − 2 = 3.
  2. Orders: 3³ = 27.
  3. Division: 27 ÷ 9 = 3.

Answer: 3

Students working together on mathematics problems
Figure 2: Group work in maths Write and evaluate two expressions with powers and brackets.

3. Common Errors

A common error is to multiply before finding powers: in 4 × 3², some learners calculate 12² = 144 instead of 4 × 9 = 36. Another is to treat 3² as 3 × 2 = 6. Always find the power first, and remember that 3² means 3 × 3.

4. Expressions in Real Life

Area problem

A square garden has sides of 7 m. A square flower bed inside it has sides of 3 m. Find the area of the garden not covered by the flower bed.

  1. Garden area: 7² = 49 m².
  2. Flower bed area: 3² = 9 m².
  3. 49 − 9 = 40.

Answer: 40 m²

New words

expression
A combination of numbers and operations.
BODMAS
The order of operations.
orders
Powers and roots in BODMAS.
brackets
Symbols ( ) that group parts of an expression.

Summary

  • Follow BODMAS: brackets, orders, division and multiplication, then addition and subtraction.
  • Find powers before multiplying or adding.
  • Expressions with powers are used in area and volume problems.

Class activity: BODMAS Challenge

  1. Each partner writes three expressions using powers and brackets.
  2. Swap and evaluate them.
  3. Check together using a calculator.

You need: Number cards, Square-grid paper, Calculator

Exercises

Answer in your exercise book first, then check your answers.

  1. 1. 3² + 1 =

    • A. 10
    • B. 7
    • C. 16
    • D. 4
    Show answer
    Answer: 109 + 1.

  2. 2. 2 × 5² =

    • A. 50
    • B. 100
    • C. 20
    • D. 25
    Show answer
    Answer: 502 × 25.

  3. 3. In BODMAS, powers come before multiplication.

    • A. True
    • B. False
    Show answer
    Answer: TrueOrders second.

  4. 4. The O in BODMAS stands for ______.

    Show answer
    Answer: ordersPowers and roots.

  5. 5. Evaluate (2 + 3)² − 4².

    Show answer
    Answer: 25 − 16 = 9Brackets, orders, subtraction.

  6. 6. Explain why 4 × 3² is not 144.

    Show answer
    Answer: Powers come first: 3² = 9, then 4 × 9 = 36; 144 comes from wrongly calculating (4 × 3)².Order of operations.

Lesson 3

Strand
Number
Sub-strand
Number Operations
Indicator
B7.1.2.3.5 – Apply the concept of powers of numbers (product of prime) to find Highest Common Factor (HCF)

Introduction

A teacher has 48 exercise books and 72 pencils. She wants to share them equally among the largest possible number of learners with nothing left over. How many learners? The answer is the highest common factor.

1. What Is the HCF?

Highest common factor (HCF): The largest number that divides two or more numbers exactly.

2. Using Prime Factors

  1. Write each number as a product of prime factors in index form.
  2. Identify the prime factors common to all the numbers.
  3. For each common prime, take the lowest power.
  4. Multiply these together to get the HCF.

Find the HCF

Find the HCF of 48 and 72.

  1. 48 = 2⁴ × 3.
  2. 72 = 2³ × 3².
  3. Common primes: 2 and 3. Lowest powers: 2³ and 3¹.
  4. HCF = 2³ × 3 = 8 × 3.

Answer: HCF = 24

A pile of exercise books and pencils
Figure 3: Sharing materials Solve a sharing problem using the HCF.

3. HCF of Three Numbers

HCF of three numbers

Find the HCF of 60, 90 and 150.

  1. 60 = 2² × 3 × 5.
  2. 90 = 2 × 3² × 5.
  3. 150 = 2 × 3 × 5².
  4. Common primes: 2, 3, 5, each with lowest power 1.

Answer: HCF = 2 × 3 × 5 = 30

4. Using the HCF in Problems

In the opening problem, the HCF of 48 and 72 is 24, so the teacher can share among 24 learners. Each learner gets 48 ÷ 24 = 2 exercise books and 72 ÷ 24 = 3 pencils. The HCF is used whenever we need the largest equal groups, the largest tile to cover a floor exactly, or the longest piece that cuts two ropes into equal lengths.

New words

HCF
The highest common factor.
common factor
A factor shared by two or more numbers.
lowest power
The smallest index of a prime among the numbers.
prime factor
A factor that is a prime number.

Summary

  • The HCF is the largest number that divides two or more numbers exactly.
  • Find it by multiplying the common prime factors with their lowest powers.
  • The HCF solves problems about the largest equal groups or pieces.

Class activity: HCF Sharing Problems

  1. Use bottle tops to model sharing 24 red and 36 blue tops into equal groups.
  2. Find the largest number of groups using the HCF.
  3. Solve two more sharing problems using prime factors.

You need: Number cards, Square-grid paper, Calculator

Exercises

Answer in your exercise book first, then check your answers.

  1. 1. The HCF of 12 and 18 is…

    • A. 6
    • B. 3
    • C. 36
    • D. 2
    Show answer
    Answer: 62 × 3.

  2. 2. 2³ × 3 and 2² × 3² have HCF…

    • A. 12
    • B. 6
    • C. 72
    • D. 24
    Show answer
    Answer: 122² × 3.

  3. 3. The HCF divides each number exactly.

    • A. True
    • B. False
    Show answer
    Answer: TrueCommon factor.

  4. 4. HCF stands for highest common ______.

    Show answer
    Answer: factorLargest divisor.

  5. 5. Find the HCF of 36 and 84 using prime factors.

    Show answer
    Answer: 36 = 2² × 3², 84 = 2² × 3 × 7; HCF = 2² × 3 = 12.Prime factors.

  6. 6. Two ropes are 84 m and 126 m long. They are cut into equal pieces as long as possible. How long is each piece?

    Show answer
    Answer: 42 m; 84 = 2² × 3 × 7, 126 = 2 × 3² × 7, HCF = 2 × 3 × 7 = 42.HCF problem.

Homework

  1. Evaluate five expressions with powers using BODMAS.
  2. Find the HCF of 54 and 90, and of 40, 60 and 100.
Evaluating Powers and Finding the HCF – Basic 7 Mathematics Notes | sirbus