Powers, roots and standard form
A power (or index) tells you how many times to multiply a number by itself. A root is the inverse — finding which number was multiplied. Standard form writes very large or very small numbers as a × 10ⁿ.
2⁵ = 2 × 2 × 2 × 2 × 2 = 32
√144 = 12 (because 12² = 144)
³√27 = 3 (because 3³ = 27)
Standard form: 3,400,000 = 3.4 × 10⁶
√144 = 12 (because 12² = 144)
³√27 = 3 (because 3³ = 27)
Standard form: 3,400,000 = 3.4 × 10⁶
n⁰ = 1
Any number to power 0
n⁻¹
= 1/n (reciprocal)
√
Square root
³√
Cube root
💡 Standard form: The number before × 10ⁿ must always be between 1 and 10. 34 × 10⁵ is not standard form — write it as 3.4 × 10⁶.
Substituting into formulae
A formula is a rule written using letters (variables) and numbers. To use a formula, you substitute (replace) the letters with the values given, then calculate.
Example: v = u + at
If u = 5, a = 3, t = 4: v = 5 + (3 × 4) = 5 + 12 = 17
If u = 5, a = 3, t = 4: v = 5 + (3 × 4) = 5 + 12 = 17
💡 Remember BODMAS: When substituting, always work out brackets and powers before multiplication and addition.
Solving linear equations
A linear equation has one unknown (usually x) and no powers. To solve it, perform the same operation on both sides until x is isolated.
Example: 3x + 7 = 22
Step 1: Subtract 7 from both sides → 3x = 15
Step 2: Divide both sides by 3 → x = 5
Step 1: Subtract 7 from both sides → 3x = 15
Step 2: Divide both sides by 3 → x = 5
💡 Check your answer: Substitute back in: 3(5) + 7 = 15 + 7 = 22 ✓
Direct and inverse proportion
Two quantities are in direct proportion if one increases, the other increases at the same rate (y = kx). They are in inverse proportion if one increases as the other decreases (y = k/x).
Direct
y = kx (doubling x doubles y)
Inverse
y = k/x (doubling x halves y)
k
Constant of proportionality
💡 Identifying type: If x doubles and y doubles → direct. If x doubles and y halves → inverse.
1Powers and roots
▼Calculate: (a) 3⁴ (b) √196 (c) ³√125
- 1(a) 3⁴ = 3 × 3 × 3 × 3 = 9 × 9 = 81
- 2(b) √196 = 14 (because 14² = 196)
- 3(c) ³√125 = 5 (because 5³ = 125)
✅ (a) 81 | (b) 14 | (c) 5
2Standard form
▼Write (a) 0.000045 and (b) 8,700,000 in standard form.
- 1(a) Move decimal to get 4.5. Moved 5 places right, so power is −5. Answer: 4.5 × 10⁻⁵
- 2(b) Move decimal to get 8.7. Moved 6 places left, so power is +6. Answer: 8.7 × 10⁶
✅ (a) 4.5 × 10⁻⁵ | (b) 8.7 × 10⁶
3Substituting into a formula
▼The formula for kinetic energy is E = ½mv². Find E when m = 4 and v = 6.
- 1Substitute values: E = ½ × 4 × 6²
- 2Powers first: 6² = 36
- 3Multiply: ½ × 4 × 36 = 2 × 36 = 72
✅ E = 72
4Solving a linear equation
▼Solve: 5x − 3 = 2x + 12
- 1Collect x terms on one side: 5x − 2x = 12 + 3
- 2Simplify: 3x = 15
- 3Divide: x = 5
- 4Check: 5(5)−3=22, 2(5)+12=22 ✓
✅ x = 5
5Direct proportion
▼y is directly proportional to x. When x = 4, y = 20. Find y when x = 11.
- 1Find constant k: y = kx → 20 = k × 4 → k = 5
- 2Formula: y = 5x
- 3When x = 11: y = 5 × 11 = 55
✅ y = 55
6Inverse proportion
▼y is inversely proportional to x. When x = 3, y = 12. Find y when x = 9.
- 1Inverse proportion: y = k/x
- 2Find k: 12 = k/3 → k = 36
- 3When x = 9: y = 36/9 = 4
✅ y = 4
Question 1
What is 2⁷?
Question 2
Write 0.0062 in standard form.
Question 3
Using v² = u² + 2as, find v when u = 0, a = 10, s = 20.
Question 3 (corrected)
Solve: 4x + 9 = 33
Question 4
y is directly proportional to x. When x = 5, y = 35. Find y when x = 8.
Question 5
Solve: 7x − 4 = 3x + 16
Question 6
y is inversely proportional to x. When x = 6, y = 8. Find y when x = 4.
Question 7
What is ³√216?
Question 8
Using E = ½mv², find E when m = 10 and v = 8.