Indices I Revision Notes

Indices, also known as exponents or powers, provide a shorthand way to express repeated multiplication of the same number. For example, 2^6 means multiplying 2 by itself 6 times: 2 × 2 × 2 × 2 × 2 × 2.

Calculating 2^6: 2 × 2 = 4, 4 × 2 = 8, 8 × 2 = 16, 16 × 2 = 32, 32 × 2 = 64. So, 2^6 = 64.

Indices simplify writing and working with large or repeated multiplications, making calculations and algebraic manipulations more manageable.

• Indices indicate how many times a base number is multiplied by itself.
• Positive indices represent repeated multiplication.
• Indices are used extensively in algebra, science, and engineering.

Positive indices show repeated multiplication, as in 3^4 = 3 × 3 × 3 × 3 = 81.

An index of zero means the value is 1, provided the base is not zero. For example, 5^0 = 1. This is because any non-zero number divided by itself equals 1, and the zero index can be understood from the division rule of indices.

Negative indices represent the reciprocal of the positive power. For example, 4^-2 means 1 divided by 4 squared: 4^-2 = 1 / 4^2 = 1 / 16 = 0.0625.

• a^0 = 1 for any a ≠ 0.
• a^-n = 1 / a^n, representing the reciprocal.
• Negative indices allow expressing division and reciprocals using indices notation.

Fractional indices represent roots. The denominator of the fraction indicates the root, and the numerator indicates the power.

For example, a^(1/2) means the square root of a, written as √a.

Similarly, a^(1/3) means the cube root of a, and a^(m/n) means the nth root of a raised to the mth power: a^(m/n) = (n√a)^m.

• a^(1/n) = nth root of a.
• Fractional indices combine roots and powers.
• Example: 9^(1/2) = √9 = 3.

The rules of indices allow us to simplify expressions involving powers with the same base.

1. Multiplying powers: a^m × a^n = a^(m+n). For example, 2^3 × 2^4 = 2^(3+4) = 2^7 = 128.

2. Dividing powers: a^m ÷ a^n = a^(m-n). For example, 2^3 ÷ 2^4 = 2^(3-4) = 2^-1 = 1/2 = 0.5. This shows how negative indices arise naturally from division rules of powers with the same base. 3. Power of a power: (a^m)^n = a^(m×n). For example, (2^3)^2 = 2^(3×2) = 2^6 = 64.

• Add indices when multiplying with the same base.
• Subtract indices when dividing with the same base.
• Multiply indices when raising a power to another power.
• These rules only apply when the base is the same.

Using the rules, we can calculate and simplify expressions efficiently.

Example 1: Calculate 2^-3 × 2^4. Using the multiplication rule, add indices: -3 + 4 = 1, so 2^-3 × 2^4 = 2^1 = 2.

Example 2: Simplify (2^3)^2. Using the power of a power rule, multiply indices: 3 × 2 = 6, so (2^3)^2 = 2^6 = 64. Example 3: Simplify 2^3 ÷ 2^4. Using the division rule, subtract indices: 3 - 4 = -1, so 2^3 ÷ 2^4 = 2^-1 = 1/2 = 0.5.

• Apply rules step-by-step to avoid mistakes.
• Check results by converting to expanded multiplication if unsure.
• Use indices to simplify complex expressions quickly.