Exponents and Powers
Class 8 · Mathematics
4 ideas in Class 8 Mathematics, explained in plain words, with games and quizzes to practise.
Powers with Negative Exponents
A negative exponent means 'one over'. So 10⁻² is 1 ÷ 10², which is 1/100, and 3⁻² is 1/9. We can see why from the pattern 100, 10, 1, then 1/10, 1/100: each step down divides by 10. The number a⁻ᵐ is the multiplicative inverse of aᵐ.
Why it matters. Negative exponents are needed to write very small numbers, like the thickness of paper. Children often think they make a number negative, which is a very common mistake.
Try it in two minutes
- Write the pattern 1000, 100, 10, 1 on paper. Ask your child what comes next if we keep dividing by 10.
- Write these as 10³, 10², 10¹, 10⁰, and ask what power matches 1/10.
- Repeat with 2³, 2², 2¹, 2⁰ and ask what 2⁻¹ and 2⁻² should be.
- Ask: is 2⁻² a negative number? Why not?
Try at home. Ask your child to write 5⁻² and 10⁻⁴ as fractions.
Laws of Exponents
Powers with the same base follow simple rules even when exponents are negative: aᵐ × aⁿ is aᵐ⁺ⁿ, aᵐ ÷ aⁿ is aᵐ⁻ⁿ, and (aᵐ)ⁿ is aᵐⁿ. Also a⁰ is 1 for any non-zero a. For example, 5⁷ × 5⁻³ is 5⁴.
Why it matters. These laws let children simplify long products and fractions quickly. A common mistake is multiplying exponents when they should be added, or the other way round.
Try it in two minutes
- Ask your child to write 2³ × 2² in full as 2 × 2 × 2 × 2 × 2 and count the twos.
- Ask them to guess the rule for multiplying powers of the same base.
- Try 2⁻¹ × 2³ as 1/2 × 8 and check that it matches 2².
- Ask: what is 7⁻² × 7²? Why is it 1?
Try at home. Write five short exponent problems on paper and ask your child to say whether to add, subtract or multiply the exponents.
Standard Form for Small and Large Numbers
Standard form writes a number as a number between 1 and 10 multiplied by a power of 10. Large numbers use positive powers, like 6200000 is 6.2 × 10⁶. Very small numbers use negative powers, like 0.00035 is 3.5 × 10⁻⁴. The power tells how many places the decimal point has moved.
Why it matters. Scientists use standard form for tiny and huge measurements. It also saves time, as nobody wants to count many zeros.
Try it in two minutes
- Write 0.0004 on paper. Ask your child to count how many places the point must move to get 4.
- Ask them to write the answer as 4 × 10 to a negative power.
- Now give 5 × 10⁻³ and ask them to write it in usual form.
- Try a large number such as 7000000 and see how the power becomes positive.
Try at home. Ask your child to write the thickness of a sheet of paper, 0.0016 cm, in standard form.
Comparing and Combining Numbers in Standard Form
To compare numbers in standard form, look at the powers first: 10⁻³ is much bigger than 10⁻⁵. To divide, divide the front numbers and subtract the powers. To add or subtract, change both numbers so that they have the same power of 10, then add the front numbers.
Why it matters. Comparing sizes of tiny things and huge things is important in science. Children often compare only the front numbers and forget about the powers.
Try it in two minutes
- Write 3 × 10⁻² and 3 × 10⁻⁵ as decimals: 0.03 and 0.00003.
- Ask your child which is bigger and by how many times.
- Write 2 × 10⁴ + 3 × 10³ as 20000 + 3000. Ask for the answer in standard form.
- Ask: why can we not just add the front numbers when the powers differ?
Try at home. Ask your child to compare 5 × 10⁻³ cm and 5 × 10⁻⁴ cm and say how many times one is the other.
What your child will practise
- Comparing and Combining Numbers in Standard Form: Candy Quick Fire (Quick Fire)
- Laws of Exponents: Number Quest (Collect the Answer)
- Laws of Exponents: Space Quest (Flip and Match)
- Powers with Negative Exponents: Mission Control Quest (Match It Up)
- Standard Form for Small and Large Numbers: Ocean Puzzle (Clue Puzzle)
- Quiz: Comparing and Combining Numbers in Standard Form (8 questions)
- Quiz: Laws of Exponents (8 questions)
- Quiz: Powers with Negative Exponents (8 questions)
- Quiz: Standard Form for Small and Large Numbers (8 questions)