What Are Indices?
Indices, also called exponents or powers, are a shorthand way to express repeated multiplication of the same number. For example, instead of writing 2 × 2 × 2, we write 2^3, which means 2 multiplied by itself 3 times.
The number being multiplied is called the base (here, 2), and the small number written above and to the right is the index or exponent (here, 3).
This notation helps simplify expressions and calculations involving repeated multiplication.
- Indices represent repeated multiplication.
- Base is the number multiplied by itself.
- Index (exponent) shows how many times the base is multiplied.
Multiplying and Dividing Powers with the Same Base
When multiplying powers that have the same base, you add the indices. This is because multiplying repeated multiplications combines the total number of times the base is multiplied.
For example, 2^3 × 2^2 means (2 × 2 × 2) × (2 × 2). Counting all the twos multiplied together gives 2 × 2 × 2 × 2 × 2 = 2^5.
Similarly, when dividing powers with the same base, you subtract the index of the denominator from the index of the numerator. This reflects cancelling out common factors in the repeated multiplication.
- Multiplication: a^m × a^n = a^(m+n)
- Division: a^m ÷ a^n = a^(m−n)
- Example: 2^3 × 2^2 = 2^(3+2) = 2^5 = 32
- Example: 5^4 ÷ 5^2 = 5^(4−2) = 5^2 = 25
The Zero Power Rule
Any non-zero number raised to the power zero equals 1. This rule is consistent with the division rule for indices.
For example, consider 3^2 ÷ 3^2. Using the division rule, this equals 3^(2−2) = 3^0. But 3^2 ÷ 3^2 is clearly 1, so 3^0 = 1.
This applies to all bases except zero, since zero to the zero power is undefined.
- a^0 = 1 for any a ≠ 0
- Example: 7^0 = 1
- This rule follows logically from the division of equal powers.
Negative Indices and Their Meaning
A negative index indicates the reciprocal of the positive power of the base. This means a^(-n) = 1/(a^n).
For example, 2^(-3) means 1 divided by 2^3, which is 1/8.
This rule extends the index laws to include negative integers and helps simplify expressions involving division and powers.
- a^(−n) = 1/(a^n)
- Example: 5^(−2) = 1/(5^2) = 1/25
- Negative indices represent reciprocal powers.
Worked Examples and Calculations
Example 1: Calculate 2 × 2, 2^3, and 2 ÷ 2 using indices.
2 × 2 is repeated multiplication of 2 twice, which is 2^2 = 4.
2^3 means 2 × 2 × 2 = 8, calculated by multiplying 2 three times together, not just 2 × 3 = 6 (which is incorrect). This shows the difference between multiplication and powers of a number, which is repeated multiplication, not repeated addition or multiplication by the index itself (exponent). 2 ÷ 2 equals 1, which can be written as 2^1 ÷ 2^1 = 2^(1−1) = 2^0 = 1, confirming the zero power rule.
- Calculate 2 × 2 = 4 (which is 2^2)
- Calculate 2^3 = 8 (2 × 2 × 2)
- Calculate 2 ÷ 2 = 1 (which is 2^0)
At-a-glance summary
Comparison table
| Step | Detail | Example | Exam Use |
|---|---|---|---|
| Express repeated multiplication | Write the base and the number of times it is multiplied | 2 × 2 × 2 = 2^3 | Use indices to simplify long multiplications |
| Multiply powers with same base | Add the indices | 2^3 × 2^2 = 2^(3+2) = 2^5 | Simplify expressions quickly |
| Divide powers with same base | Subtract the indices | 5^4 ÷ 5^2 = 5^(4−2) = 5^2 | Avoid long division calculations |
| Zero power | Any non-zero base to zero power equals 1 | 7^0 = 1 | Recognise zero powers in expressions |
| Negative power | Reciprocal of positive power | 3^(−2) = 1/(3^2) = 1/9 | Simplify expressions with negative indices |
Active recall
Quick check
1. What does 3^4 represent?
3^4 means 3 multiplied by itself 4 times: 3 × 3 × 3 × 3 = 81.
2. Calculate 5^3 × 5^2.
Add the indices: 5^(3+2) = 5^5 = 3125.
3. What is the value of 4^0?
4^0 = 1, since any non-zero number to the zero power equals 1.
4. Express 2^(−3) as a fraction.
2^(−3) = 1/(2^3) = 1/8.
5. Calculate 7^5 ÷ 7^2.
Subtract the indices: 7^(5−2) = 7^3 = 343.