Indices (Powers and Exponents)
Section: Number and Calculation | Syllabus: Cambridge Lower Secondary Checkpoint Mathematics (0862)
What are Indices?
An index (plural: indices), also called a power or exponent, is a shorthand for repeated multiplication. Instead of writing 2 × 2 × 2, you can write 2³.
- In an, a is the base (the number being multiplied) and n is the index (how many times it's multiplied by itself).
- an means a × a × a × ... repeated n times.
| Index Form | Expanded Form | Value |
|---|---|---|
| 23 | 2 × 2 × 2 | 8 |
| 52 | 5 × 5 | 25 |
| 104 | 10 × 10 × 10 × 10 | 10,000 |
| 31 | 3 | 3 |
The index tells you how many times the base is written down and multiplied
Laws of Indices
The five laws below let you simplify expressions without expanding every power - as long as the bases involved are identical.
Law 1: Multiplication (Same Base)
am × an = am+n
When multiplying powers with the same base, add the indices.
-
Question: Simplify 25 × 23.
- Step 1: The bases match (both 2), so add the indices: 25 × 23 = 25+3
- Step 2: 25+3 = 28
- Answer: 28 = 256
Law 2: Division (Same Base)
am ÷ an = am-n
When dividing powers with the same base, subtract the indices.
-
Question: Simplify 57 ÷ 53.
- Step 1: The bases match (both 5), so subtract the indices: 57 ÷ 53 = 57-3
- Step 2: 57-3 = 54
- Answer: 54 = 625
Law 3: Power of a Power
(am)n = am×n
When raising a power to another power, multiply the indices.
-
Question: Simplify (34)2.
- Step 1: Multiply the indices: (34)2 = 34×2
- Step 2: 34×2 = 38
- Answer: 38 = 6561
Law 4: Power of a Product
(ab)n = an × bn
Distribute the power to each factor inside the brackets. For example: (2 × 3)3 = 23 × 33 = 8 × 27 = 216, which matches 63 = 216 found directly.
Law 5: Power of a Quotient
(a/b)n = an ÷ bn
Distribute the power to the numerator and denominator. For example: (4/2)2 = 42 ÷ 22 = 16 ÷ 4 = 4, which matches 22 = 4 found directly.
Special Index Rules
Zero Index
a0 = 1 (where a ≠ 0)
- Any number (except zero) raised to the power of 0 equals 1.
- Examples: 50 = 1, 1000 = 1, (−7)0 = 1
Negative Indices
a-n = 1/an
- A negative index means "take the reciprocal" - it does not make the value negative.
- Examples: 2-3 = 1/23 = 1/8, 5-2 = 1/52 = 1/25, 10-1 = 1/10 = 0.1
- To move a power from the numerator to the denominator (or vice versa), change the sign of its index: 3/x2 = 3x-2
-
Question: Simplify x5 × x-2.
- Step 1: Same base, so add the indices: x5 × x-2 = x5+(-2)
- Answer: x5+(-2) = x3
Fractional Indices (Roots)
a1/n = n√a
am/n = (n√a)m or n√(am)
| Index Form | Root Form | Example |
|---|---|---|
| a1/2 | √a (square root) | 161/2 = √16 = 4 |
| a1/3 | ∛a (cube root) | 271/3 = ∛27 = 3 |
| a1/4 | 4√a (fourth root) | 811/4 = 4√81 = 3 |
| a2/3 | (∛a)2 | 82/3 = (∛8)2 = 22 = 4 |
| a3/2 | (√a)3 | 43/2 = (√4)3 = 23 = 8 |
-
Question: Evaluate 163/4.
- Method 1: Take the root first: 163/4 = (4√16)3 = 23 = 8
- Method 2: Raise to the power first: 163/4 = 4√(163) = 4√4096 = 8
- Answer: 163/4 = 8 (both methods agree - Method 1 is usually easier)
-
Question: Evaluate 8-2/3.
- Step 1: Deal with the negative index first: 8-2/3 = 1/82/3
- Step 2: Evaluate the fractional power: 82/3 = (∛8)2 = 22 = 4
- Answer: 1/4
Quick Reference Table
| Rule | Formula | Example |
|---|---|---|
| Multiplication | am × an = am+n | x3 × x5 = x8 |
| Division | am ÷ an = am-n | y7 ÷ y2 = y5 |
| Power of power | (am)n = amn | (z2)3 = z6 |
| Zero index | a0 = 1 | 70 = 1 |
| Negative index | a-n = 1/an | 3-2 = 1/9 |
| Fractional index | a1/n = n√a | 251/2 = 5 |
| Fractional power | am/n = (n√a)m | 272/3 = (∛27)2 = 9 |
Exam Tips
Remember
- Same base? → Use the laws of indices
- Multiplying? → ADD the powers
- Dividing? → SUBTRACT the powers
- Power of power? → MULTIPLY the powers
- Negative power? → Take the reciprocal
- Fractional power? → Use roots
Practice Strategy
Interactive revision notes, videos and practice questions load below.