Indices II Revision Notes

Indices, also known as powers or exponents, express how many times a number (called the base) is multiplied by itself. For example, 3^4 means 3 × 3 × 3 × 3, which equals 81.

Indices provide a concise way to write repeated multiplication and are fundamental in algebra and higher mathematics.

• The base is the number being multiplied.
• The index (or exponent) tells how many times the base is used as a factor.
• Indices simplify notation and calculations involving repeated multiplication.

There are four key rules that govern how indices behave when multiplying, dividing, or raising powers to powers. These rules apply when the bases are the same and are essential for simplifying expressions efficiently.

• Multiplying powers: a^m × a^n = a^(m + n)
• Dividing powers: a^m ÷ a^n = a^(m - n)
• Power of a power: (a^m)^n = a^(m × n)
• Zero power: a^0 = 1, provided a ≠ 0

A zero index means the base is raised to the power zero, which always equals 1 (except when the base is zero, which is undefined). This is because any number divided by itself equals 1, and the zero power can be derived from the division rule.

Negative indices represent the reciprocal of the base raised to the corresponding positive power. For example, a^(-n) = 1 / a^n. This allows expressions with negative powers to be rewritten as fractions, which is useful for simplification.

• Zero index means the value is 1 (a^0 = 1).
• Negative index means reciprocal: a^(-n) = 1 / a^n.
• These rules extend the laws of indices to all integer powers.

To simplify expressions involving indices, apply the rules step-by-step, combining like bases by adding or subtracting exponents as appropriate.

When expressions include multiplication and division, rewrite all terms with indices and then use the rules to simplify.

Be careful to handle coefficients (numbers without indices) separately from the powers, and apply the rules only to terms with the same base.

• Identify bases and their indices.
• Use multiplication and division rules to combine powers.
• Rewrite negative indices as reciprocals if needed.
• Simplify coefficients separately from powers.
• Check the final expression for any further simplification.

Step 1: Write all terms clearly and separate coefficients and variables with indices:

3^2 × 5 × x^1 × 12 × a^5 ÷ 3 × a^(-2) × 6 × x^7 × y^4 × 5 × x^(-5) × y^1

Step 2: Combine all coefficients (numbers): 3^2 = 9, so coefficients are 9 × 5 × 12 ÷ 3 × 6 × 5. Calculate numerator and denominator separately: numerator = 9 × 5 × 12 = 540; denominator = 3 × 6 = 18; then multiply by 5 in numerator: 540 × 5 = 2700; divide by 18: 2700 ÷ 18 = 150 (final coefficient). Note: The 5 at the end is multiplied, so included in numerator multiplication before division by denominator terms only 3 and 6. Alternatively, multiply all numerator coefficients and divide by denominator coefficients: numerator coefficients: 9 × 5 × 12 × 5 = 2700; denominator coefficients: 3 × 6 = 18; 2700 ÷ 18 = 150. This confirms the coefficient is 150. Step 3: Combine powers with the same base using index rules: - For a: a^5 ÷ a^(-2) = a^(5 - (-2)) = a^(5 + 2) = a^7 - For x: x^1 × x^7 × x^(-5) = x^(1 + 7 - 5) = x^3 - For y: y^4 × y^1 = y^(4 + 1) = y^5 Step 4: Write the simplified expression: 150 × a^7 × x^3 × y^5 Step 5: Final answer: 150a^7x^3y^5 This example shows how to carefully apply the rules of indices and handle coefficients to simplify a complex expression.

• Separate coefficients and variables.
• Apply index rules to each base.
• Calculate coefficients separately.
• Combine results for the final simplified expression.