EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Algebraic indices
Algebra · Lesson 1 of 7
From the Number Chapter
Answer each one, then check.
1. Write 2⁵ × 2³ as a single power of 2.
2⁸
2. Write (3²)⁴ as a single power of 3.
3⁸
3. What is 5⁰?
1
4. What is 3⁻²?
1/9
5. Simplify 3a × 4b.
12ab
Learning Objectives
1. Use the index laws with letters.
2. Simplify expressions with numbers and several letters.
3. Raise a bracket to a power, such as (2x³)⁴.
4. Use negative and fractional indices in algebra. (Higher)
5. Solve equations with unknown powers, such as 9^x = 27^(x−1). (Higher)
The Index Laws with Letters
Exactly the laws from the Number chapter - the base is now a letter.
|
Law |
In words |
Example |
|---|---|---|
|
Multiplying |
Add the indices |
x⁵ × x² = x⁷ |
|
Dividing |
Subtract the indices |
y⁸ ÷ y² = y⁶ |
|
Power of a power |
Multiply the indices |
(z³)⁵ = z¹⁵ |
|
Zero index |
Anything (except 0) to the power 0 is 1 |
x⁰ = 1 |
|
A letter on its own |
Has an index of 1 |
x × x⁴ = x¹ × x⁴ = x⁵ |
Multiplying Terms
|
Simplify 3x²y⁴ × 5x³y |
1. Multiply the numbers
3 × 5 = 15
2. Multiply the x terms: add the indices
x² × x³ = x⁵
3. Multiply the y terms: y is y¹
y⁴ × y¹ = y⁵
4. Put them together
15x⁵y⁵
Answer: 15x⁵y⁵
Dividing Terms
|
Simplify 24a⁷b³ ÷ 8a²b³ |
1. Divide the numbers
24 ÷ 8 = 3
2. Divide the a terms: subtract the indices
a⁷ ÷ a² = a⁵
3. Divide the b terms
b³ ÷ b³ = b⁰ = 1
4. Put them together
3 × a⁵ × 1
Answer: 3a⁵
A Bracket to a Power
|
Simplify (3x²y⁵)³ |
1. Everything inside the bracket is raised to the power 3
3³ × (x²)³ × (y⁵)³
2. The number
3³ = 27
3. Power of a power: multiply the indices
(x²)³ = x⁶ and (y⁵)³ = y¹⁵
Answer: 27x⁶y¹⁵
Why a Bracket Cubed Cubes the Number Too
|
The power on a bracket applies to everything inside it - the number as well as the letter. A cube with edges of 2x has a volume of 2x × 2x × 2x = 8x³. Writing 2x³ would mean only the x was cubed. |
(2x)³ = 2x × 2x × 2x = 8x³, not 2x³. |
Mistakes to Avoid
|
WRONG |
RIGHT |
|
▸ (2x)³ = 2x³ ▸ 3x² × 2x = 6x² ▸ x² + x³ = x⁵ ▸ (x³)² = x⁵ ▸ 12x⁶ ÷ 4x² = 3x³ |
▸ (2x)³ = 8x³ - cube the 2 as well. ▸ 3x² × 2x = 6x³ - x is x¹. ▸ x² + x³ cannot be simplified: they are not like terms. ▸ (x³)² = x⁶ - multiply the indices. ▸ 12x⁶ ÷ 4x² = 3x⁴ - subtract the indices. |
PART TWO · HIGHER
Negative and Fractional Indices
The rest of the index rules, now with letters.
Negative and Fractional Powers of Letters
The meanings are the same as for numbers.
▸ Negative powers. x⁻¹ = 1/x and x⁻³ = 1/x³. So 1/x⁴ can be written as x⁻⁴.
▸ Fractional powers. x^(1/2) = √x and x^(1/3) = ∛x. So 1/(√x) = x^(−1/2).
▸ "Write in the form xⁿ". Turn roots and fractions into single powers: 1/x² = x⁻², √x³ = x^(3/2).
▸ Brackets. Raise each part inside the bracket to the power: (16x⁸)^(1/2) = 16^(1/2) × x⁴ = 4x⁴.
A Negative Fractional Power
|
Simplify (27x⁶y⁻³)^(−1/3) |
1. Raise each part to the power −⅓
27^(−1/3) × (x⁶)^(−1/3) × (y⁻³)^(−1/3)
2. The number: flip and cube root
27^(−1/3) = ⅓
3. The x: multiply the indices
6 × (−⅓) = −2, so x⁻² = 1/x²
4. The y: multiply the indices
−3 × (−⅓) = 1, so y¹ = y
5. Put them together
⅓ × 1/x² × y
Answer: y/3x²
Solving an Equation with Powers
|
Solve 2^(x+1) = 8^x |
1. Write both sides with the same base
8 = 2³, so 8^x = 2^(3x)
2. So
2^(x+1) = 2^(3x)
3. Same base, so the indices are equal
x + 1 = 3x
4. Solve
1 = 2x
Answer: x = ½
Key Terms
|
Index (power) The small number showing how many times the base is multiplied by itself. |
Coefficient The number in front of a letter, e.g. 5 in 5x³. |
|
Like terms Terms with exactly the same letters and powers, e.g. 3x² and 7x². |
Simplify Write an expression in its shortest form. |
|
Negative index (Higher) x⁻ⁿ = 1/xⁿ. |
Fractional index (Higher) x^(1/n) = ⁿ√x. |
Your Task: Index Pyramids
12 minutes
|
Simplify each. (a) a⁴ × a⁶ (b) 6m⁵ × 2m (c) 18p⁸q³ ÷ 6p²q (d) (4x³)² (e) (2a²b)³ × 3ab⁴ (f) (x⁷ × x²)/x⁵. Higher: (g) (9x⁴)^(1/2) (h) (8y⁶)^(−1/3) (i) solve 4^x = 2^(x+3). 1. Numbers first. 2. Then each letter in turn. 3. Check every bracket's power applies to everything inside. |
A good answer shows: (a) a¹⁰ (b) 12m⁶ (c) 3p⁶q² (d) 16x⁶ (e) 8a⁶b³ × 3ab⁴ = 24a⁷b⁷ (f) x⁴ (g) 3x² (h) 1/2y² (i) 2^(2x) = 2^(x+3), so x = 3.
Can I...?
☐ Multiply terms using the index laws.
☐ Divide terms using the index laws.
☐ Raise a power to a power.
☐ Raise a bracket to a power.
☐ Use x⁰ = 1.
☐ Use negative powers of letters. (Higher)
☐ Use fractional powers of letters. (Higher)
☐ Solve equations with unknown powers. (Higher)
Summary
✓ Multiply: add indices. Divide: subtract indices. Power of a power: multiply indices.
✓ A power on a bracket applies to everything inside, including the number.
✓ Only like terms can be added; x² + x³ does not simplify.
✓ (Higher) x⁻ⁿ = 1/xⁿ and x^(1/n) = ⁿ√x.
|
EXAM FOCUS Simplify (2a³b)⁵ (2 marks) Work through the numbers first, then each letter in alphabetical order. The two marks are usually one for the number and one for the letters. |