Understanding Indices and Their Meaning
Indices, also called exponents or powers, indicate how many times a base number is multiplied by itself. For example, 3^4 means 3 × 3 × 3 × 3, which equals 81.
The base is the number being multiplied, and the index (or exponent) tells how many times to multiply the base.
Indices simplify writing repeated multiplication and help in algebraic manipulation.
- 3^4 = 3 × 3 × 3 × 3 = 81
- Base: 3, Index: 4
- Indices represent repeated multiplication
Key Rules for Positive and Zero Indices
When multiplying powers with the same base, add their indices. For example, x^2 × x^3 = x^(2+3) = x^5.
When dividing powers with the same base, subtract the indices: x^5 ÷ x^2 = x^(5−2) = x^3.
Raising a power to another power multiplies the indices: (x^2)^3 = x^(2×3) = x^6. This rule also applies to numerical bases: (2^3)^2 = 2^(3×2) = 2^6 = 64.
- Product rule: a^m × a^n = a^(m+n)
- Quotient rule: a^m ÷ a^n = a^(m−n)
- Power of a power: (a^m)^n = a^(m×n)
- Zero power rule: a^0 = 1 for a ≠ 0
Interpreting Zero and Negative Indices
Any nonzero number raised to the zero power equals 1. For example, 5^0 = 1. This is consistent with the quotient rule: dividing a^m by itself gives a^(m−m) = a^0 = 1.
Negative indices represent the reciprocal of the positive power. For example, a^(−n) = 1/a^n. So, 2^(−3) = 1/(2^3) = 1/8.
This allows expressions with negative powers to be rewritten as fractions, which is useful for simplification and solving equations.
- Zero power: a^0 = 1 (a ≠ 0)
- Negative power: a^(−n) = 1/a^n
- Example: 3^(−2) = 1/9
Fractional Indices and Their Meaning
Fractional indices represent roots combined with powers. The index m/n means the nth root of the base raised to the mth power: a^(m/n) = (ⁿ√a)^m = ⁿ√(a^m).
For example, 9^(1/2) means the square root of 9, which is 3. Similarly, 8^(2/3) means the cube root of 8 squared: (³√8)^2 = 2^2 = 4.
Fractional indices allow us to express roots and powers in a single notation, simplifying algebraic manipulation.
- Fractional index: a^(m/n) = ⁿ√(a^m)
- Example: 16^(3/4) = (⁴√16)^3 = 2^3 = 8
- Square root is a special case: a^(1/2) = √a
Simplifying Expressions Using Index Rules
To simplify expressions with indices, apply the rules step-by-step, combining like bases and carefully handling negative and fractional powers.
Example: Simplify x^2 × x^3 × x^2 × x^(−2) × x^5. Add the indices: 2 + 3 + 2 − 2 + 5 = 10, so the expression simplifies to x^10.
Example: Simplify 2^3 − 4^1 × x. Calculate powers first: 2^3 = 8 and 4^1 = 4, so the expression is 8 − 4x. This shows the importance of order of operations and careful substitution.
- Combine powers with the same base by adding or subtracting indices
- Calculate powers before addition or subtraction
- Keep track of negative indices as reciprocals
Solving Equations Involving Indices
To solve equations like x × 3^2 = 2^(x+1), express all terms with indices and isolate the variable.
Example: Solve x × 3^2 = 2^(x+1). First, calculate 3^2 = 9, so the equation is 9x = 2^(x+1). This equation involves different bases and may require trial or estimation since logarithms are not required.
Example: Solve x × 5 = 25. Divide both sides by 5: x = 25 ÷ 5 = 5. This is a straightforward linear equation involving indices as coefficients rather than powers.
- Express terms with indices clearly
- Isolate the variable using algebraic operations
- Use substitution or trial if bases differ and logarithms are not allowed
At-a-glance summary
Comparison table
| Step | Detail | Example | Exam Use |
|---|---|---|---|
| Product rule | Multiply powers with the same base by adding indices | x^2 × x^3 = x^5 | Combine powers quickly to simplify expressions |
| Quotient rule | Divide powers with the same base by subtracting indices | x^5 ÷ x^2 = x^3 | Simplify division of powers efficiently |
| Power of a power | Raise a power to another power by multiplying indices | (x^2)^3 = x^6 | Simplify nested powers accurately |
| Zero power | Any nonzero base raised to zero equals one | 5^0 = 1 | Recognise and simplify zero powers correctly |
| Negative power | Negative index means reciprocal of positive power | 2^(−3) = 1/8 | Rewrite negative powers as fractions for simplification |
| Fractional power | Fractional index represents roots and powers | 9^(1/2) = 3 | Express roots using indices for algebraic manipulation |
Active recall
Quick check
1. What is the value of 4^0?
4^0 = 1 because any nonzero number raised to the zero power equals 1.
2. Simplify x^3 × x^(−1).
x^3 × x^(−1) = x^(3−1) = x^2 by subtracting indices for multiplication.
3. Express 27^(2/3) as a root and power, then calculate its value.
27^(2/3) = (³√27)^2 = 3^2 = 9 because the denominator 3 indicates cube root and numerator 2 indicates square.
4. Simplify (x^4)^3.
(x^4)^3 = x^(4×3) = x^12 by multiplying indices when raising a power to a power.