Basic 8 · Mathematics · Indices
Applying Laws of Indices, the Order of Operations and Exponential Equations
In this topic you will apply the laws of indices to simplify and evaluate expressions, use the order of operations with powers, solve simple exponential equations and solve real-life problems involving powers.
By the end of these notes you should be able to:
- Simplify expressions using the laws of indices
- Evaluate expressions using the order of operations (PEDMAS)
- Solve exponential equations and real-life problems
What you already know
- Laws of indices
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
- Indices
- Indicator
- B8.1.2.3.2 – Apply the laws of indices to simplify and evaluate numbers involving powers of numbers. (PEDMAS)
Introduction
With the laws of indices, an expression that looks complicated can become a single small number.
1. Simplifying with Numbers
Simplify
Simplify (2⁵ × 2³) ÷ 2⁶.
- 2⁵ × 2³ = 2⁸.
- 2⁸ ÷ 2⁶ = 2².
Answer: 4
Different forms of the base
Simplify 8² × 4³ ÷ 2⁵.
- Write each base as a power of 2: 8 = 2³, 4 = 2².
- (2³)² × (2²)³ ÷ 2⁵ = 2⁶ × 2⁶ ÷ 2⁵.
- 2¹² ÷ 2⁵ = 2⁷.
Answer: 128
2. Simplifying with Letters
The same laws work with letters: x³ × x⁴ = x⁷; y⁸ ÷ y³ = y⁵; (a²)⁴ = a⁸. When numbers are attached, multiply the numbers and add the indices: 3x² × 4x³ = 12x⁵. Divide similarly: 10y⁶ ÷ 2y² = 5y⁴.
3. Fractional Thinking
Sometimes a simplified expression gives a fraction. 3² ÷ 3⁵ = 3⁻³ = 1/27. Negative indices remind us that dividing by a larger power gives a value less than 1. Write the final answer as a fraction or a decimal, as required.
4. Checking by Calculating
You can check by evaluating both sides. 2³ × 2⁴ = 8 × 16 = 128, and 2⁷ = 128. Doing this a few times builds trust in the laws and helps you to spot mistakes such as multiplying indices when you should add them.
New words
- simplify
- To write in a shorter form.
- expression
- A mathematical phrase.
- coefficient
- The number multiplied by a letter.
- reciprocal
- One divided by a number.
Summary
- Write all bases in the same form, then apply the laws.
- With letters, multiply numbers and add indices.
- Check by evaluating both sides.
Class activity: Simplify Challenge
- Write five expressions with indices.
- Swap and simplify.
- Check by calculating both sides.
You need: Number cards, Square-grid paper, Calculator
Exercises
Answer in your exercise book first, then check your answers.
1. x² × x⁵ =
- A. x⁷
- B. x¹⁰
- C. 2x⁷
- D. x³
Show answer
Answer: x⁷Add indices.2. 2x³ × 5x² =
- A. 10x⁵
- B. 7x⁵
- C. 10x⁶
- D. 7x⁶
Show answer
Answer: 10x⁵Multiply and add.3. 3² ÷ 3⁵ = 1/27.
- A. True
- B. False
Show answer
Answer: True3⁻³.4. (a³)² = a to the power ______.
Show answer
Answer: 6Multiply.5. Simplify 9² × 3³ ÷ 3⁵.
Show answer
Answer: 3⁴ × 3³ ÷ 3⁵ = 3² = 9Same base.6. Simplify 12y⁷ ÷ 4y³.
Show answer
Answer: 3y⁴Divide numbers and subtract indices.
Lesson 2
- Strand
- Number
- Sub-strand
- Indices
- Indicator
- B8.1.2.3.2 – Apply the laws of indices to simplify and evaluate numbers involving powers of numbers. (PEDMAS)
Introduction
2 + 3² × 4 can give several answers if you work in the wrong order. Mathematicians agree on one order.
1. PEDMAS and BODMAS
| Step | PEDMAS | BODMAS |
|---|---|---|
| 1 | parentheses | brackets |
| 2 | exponents | orders |
| 3 | division and multiplication, left to right | division and multiplication, left to right |
| 4 | addition and subtraction, left to right | addition and subtraction, left to right |
PEDMAS and BODMAS are two names for the same order. Parentheses are brackets, and exponents are orders (powers).
2. Worked Examples
Evaluate
Find 2 + 3² × 4.
- Exponent: 3² = 9.
- Multiply: 9 × 4 = 36.
- Add: 2 + 36.
Answer: 38
With brackets
Find (5 + 2)² − 3 × 2³.
- Brackets: 7² = 49.
- Exponent: 2³ = 8.
- Multiply: 3 × 8 = 24.
- Subtract: 49 − 24.
Answer: 25
3. Left to Right
Division and multiplication have equal rank, so work from left to right: 24 ÷ 4 × 2 = 6 × 2 = 12, not 24 ÷ 8 = 3. Addition and subtraction are the same: 10 − 3 + 2 = 9, not 5.
4. Calculators and Brackets
Scientific calculators follow this order automatically, but simple calculators may not. Test yours with 2 + 3 × 4. A scientific calculator gives 14; a basic one may give 20. Use brackets on the calculator when entering long expressions.
New words
- PEDMAS
- The order of operations.
- exponent
- A power.
- parentheses
- Brackets.
- rank
- Level of priority.
Summary
- Work brackets, powers, division and multiplication, then addition and subtraction.
- Operations of equal rank go from left to right.
- Basic calculators may not follow this order.
Class activity: Order Puzzle
- Insert brackets in 2 + 3 × 4² to make different answers.
- Evaluate each.
- Find the largest possible answer.
You need: Number cards, Square-grid paper, Calculator
Exercises
Answer in your exercise book first, then check your answers.
1. 2 + 3 × 4 =
- A. 14
- B. 20
- C. 24
- D. 9
Show answer
Answer: 14Multiply first.2. (2 + 3)² =
- A. 25
- B. 13
- C. 10
- D. 29
Show answer
Answer: 25Brackets first.3. Powers come before multiplication.
- A. True
- B. False
Show answer
Answer: TrueOrder.4. 24 ÷ 4 × 2 = ______.
Show answer
Answer: 12Left to right.5. Evaluate 3 × 2³ + 4².
Show answer
Answer: 24 + 16 = 40Order.6. Evaluate (8 − 2)² ÷ 4 + 5 × 2.
Show answer
Answer: 36 ÷ 4 + 10 = 19Order.
Lesson 3
- Strand
- Number
- Sub-strand
- Indices
- Indicator
- B8.1.2.3.3-4 – Solve exponential equations and Solve real life problems involving powers of natural numbers
Introduction
Bacteria can double every hour. Money can grow with interest. Powers describe such rapid growth.
1. Exponential Equations
Exponential equation: An equation in which the unknown is in the index.
Solve
Solve 2ˣ = 32.
- Write 32 as a power of 2: 32 = 2⁵.
- 2ˣ = 2⁵.
Answer: x = 5
Solve
Solve 3²ˣ = 81.
- 81 = 3⁴.
- So 2x = 4.
Answer: x = 2
The key idea is that if the bases are equal, then the indices must be equal.
2. Doubling Problems
Growth
A virus message is forwarded so that the number of people who receive it doubles each day. If 3 people receive it on day 1, how many receive it on day 6?
- Each day multiplies by 2.
- Day 6: 3 × 2⁵.
- 3 × 32.
Answer: 96 people
3. Powers in Science and Finance
Populations, bacteria and viruses can grow exponentially. Radioactive substances decay by halving. Money with compound interest also grows by powers: GH₵1,000 at 10% a year becomes 1,000 × 1.1² = GH₵1,210 after two years. Powers help us to predict these changes.
4. Fast Growth
Exponential growth surprises people. If a saving doubled every day from 1 pesewa, it would be over GH₵5 million after 30 days, because 2²⁹ is about 537 million pesewas. This shows how quickly powers can grow, and why rumours and diseases can spread so fast.
New words
- exponential equation
- An equation with the unknown in the index.
- exponential growth
- Growth by repeated multiplication.
- compound interest
- Interest earned on interest.
- decay
- Reduction by repeated division.
Summary
- Solve exponential equations by writing both sides with the same base.
- Doubling and halving are described by powers.
- Exponential growth is very fast and appears in science and finance.
Class activity: Paper Folding
- Fold a sheet of paper in half repeatedly.
- Count the layers after each fold.
- Write the number of layers as a power of 2.
You need: Number cards, Square-grid paper, Calculator
Exercises
Answer in your exercise book first, then check your answers.
1. If 2ˣ = 8, x =
- A. 3
- B. 4
- C. 2
- D. 8
Show answer
Answer: 32³ = 8.2. If 5ˣ⁺¹ = 25, x =
- A. 1
- B. 2
- C. 0
- D. 5
Show answer
Answer: 1x + 1 = 2.3. If the bases are equal, the indices are equal.
- A. True
- B. False
Show answer
Answer: TrueRule.4. If 3ˣ = 27, then x = ______.
Show answer
Answer: 33³.5. Solve 2ˣ = 64.
Show answer
Answer: x = 62⁶.6. A population of 1,000 bacteria doubles every hour. How many after 5 hours?
Show answer
Answer: 32,0001,000 × 2⁵.
Homework
- Solve five exponential equations.
- Write a real-life doubling problem and solve it.