Power Play
Class 8 · Mathematics
5 ideas in Class 8 Mathematics, explained in plain words, with games and quizzes to practise.
Exponents and Exponential Form
An exponent tells how many times a base is multiplied by itself. In 5⁴ the base is 5 and the exponent is 4, so 5⁴ means 5 × 5 × 5 × 5, which is 625. Repeated addition is different: 4 + 4 + 4 is 3 × 4, but 4 × 4 × 4 is 4³, which is 64. Prime factors can be written in this form too, such as 72 is 2³ × 3².
Why it matters. Exponents let us write very big numbers in a short way and spot patterns in doubling and tripling. Children who mix up 4³ with 4 × 3 will make mistakes in every later topic that uses powers.
Try it in two minutes
- Ask your child to write 2 × 2 × 2 × 2 × 2 in short form and say what the base and exponent are.
- Ask for the value of 2⁵ and then for 2 × 5. Why are they different?
- Fold a newspaper in half again and again. Each fold doubles the thickness, so after 3 folds the thickness is 2³ times as much.
Try at home. Ask your child to write the number 360 as a product of prime factors in exponent form.
Rules for Multiplying, Dividing and Raising Powers
When powers share the same base, multiplying adds the exponents: 2³ × 2⁴ is 2⁷. Dividing subtracts them: 2⁷ ÷ 2⁴ is 2³. A power of a power multiplies them: (2³)⁴ is 2¹². Also, when the exponent is the same, bases can be combined: 2⁵ × 5⁵ is 10⁵.
Why it matters. These rules turn long calculations into short ones and are the base of algebra. Children who add the bases or multiply the exponents wrongly will get answers that are far off.
Try it in two minutes
- Write 2³ × 2² as (2 × 2 × 2) × (2 × 2) and count the 2s. Ask what rule that shows.
- Ask for 2⁵ ÷ 2² by cancelling twos. Ask what happened to the exponents.
- Ask why (2²)³ is 2⁶ by writing it as (2 × 2) × (2 × 2) × (2 × 2).
Try at home. Ask your child to write 2¹² as a power of a power in two different ways.
Zero and Negative Exponents
Halving a line again and again shows what powers mean beyond counting numbers. 2⁴ ÷ 2⁴ is 1, so 2⁰ is 1. 2⁴ ÷ 2⁵ is 1/2, so 2⁻¹ is 1/2. In general n⁻ᵃ is 1 divided by nᵃ, so 10⁻³ is 1/1000. A negative exponent does not make the number negative; it makes it small.
Why it matters. Negative exponents appear when we write very small numbers and when we compare sizes. Children often think 2⁻³ is −8, which leads to wrong answers.
Try it in two minutes
- Draw a line 16 cm long and halve it: 8, 4, 2, 1 cm. Write these as 2³, 2², 2¹, 2⁰.
- Halve once more and ask what length that is (1/2 cm) and how it can be written (2⁻¹).
- Ask what 10⁻² might be if 10² is 100.
Try at home. Ask your child to write 1/81 as a power of 3 and a power of 9.
Powers of 10 and Scientific Notation
Place values are powers of 10: 4506 is 4 × 10³ + 5 × 10² + 0 × 10¹ + 6 × 10⁰. Decimals use negative powers, so the 7 in 62.17 stands for 7 × 10⁻². Scientific (standard) form writes any number as x × 10ʸ where x is at least 1 and less than 10, such as 83,00,000 is 8.3 × 10⁶. The exponent shows how big the number is.
Why it matters. Very large numbers such as the distance to the Sun are hard to read with many zeros. Standard form makes them short and easy to compare.
Try it in two minutes
- Write 3482 in expanded form using powers of 10 together with your child.
- Pick a big number like 25,00,000 and ask your child to write it as 2.5 × 10 raised to some power. Count the zeros and shift together.
- Ask which is bigger: 9 × 10⁴ or 1 × 10⁵? Why does the exponent matter more?
Try at home. Ask your child to find the populations of three cities and write each in standard form.
Counting with Powers and Fast Growth
To count the ways to choose, multiply the options at each step. With 4 shirts and 3 caps there are 4 × 3 ways. A lock with 5 slots, each with 10 digits, has 10 × 10 × 10 × 10 × 10, which is 10⁵ codes. Growth that adds the same amount each time is linear. Growth that multiplies by the same number each time is exponential, and it becomes huge very quickly.
Why it matters. Counting with powers explains why long passwords are safer, and exponential growth explains things like folded paper and spreading plants. Without this idea children underestimate how quickly doubling grows.
Try it in two minutes
- Ask how many two-digit lock codes there are if each digit can be 0 to 9 (100). Then three digits (1000).
- Ask what changes when one more digit is added. (It multiplies by 10.)
- Compare adding 2 each day with doubling each day for 10 days and see which is larger.
Try at home. Ask your child how many different codes a lock with 4 slots has if each slot can be any letter from A to Z.
What your child will practise
- Counting with Powers and Fast Growth: Desert Quick Fire (Quick Fire)
- Exponents and Exponential Form: Number Quest (Match It Up)
- Powers of 10 and Scientific Notation: Jungle Puzzle (Clue Puzzle)
- Rules for Multiplying, Dividing and Raising Powers: Lab Explorer Quest (Collect the Answer)
- Zero and Negative Exponents: Jungle Quest (Flip and Match)
- Quiz: Counting with Powers and Fast Growth (8 questions)
- Quiz: Exponents and Exponential Form (8 questions)
- Quiz: Powers of 10 and Scientific Notation (8 questions)
- Quiz: Rules for Multiplying, Dividing and Raising Powers (8 questions)