Indices (laws of indices) — KCSE Mathematics

In mathematics, understanding indices is crucial for simplifying expressions. The laws of indices allow us to manipulate powers effectively.

Key Laws of Indices:

• Product of Powers: When multiplying like bases, add the indices: a^m × a^n = a^(m+n).
• Quotient of Powers: When dividing like bases, subtract the indices: a^m ÷ a^n = a^(m-n).
• Power of a Power: When raising a power to another power, multiply the indices: (a^m)^n = a^(m×n).
• Negative Indices: A negative index indicates a reciprocal: a^(-n) = 1/(a^n).
• Fractional Indices: A fractional index represents roots: a^(1/n) = n√a.

Example: Simplify the expression (2^3 × 2^(-1) ÷ 4^(1/2)).

• Step 1: Apply the product of powers: 2^(3-1) = 2^2.
• Step 2: Simplify 4^(1/2) to 2: 2^2 ÷ 2 = 2.
• Final answer: 2.

To solve equations involving indices, you need to express both sides as powers of the same base. This allows you to equate the exponents. Here are the steps to follow:

1. Identify the bases: Look for a common base for both sides of the equation.
2. Rewrite the equation: Express each side of the equation using the identified base.
3. Equate the exponents: Once the bases are the same, set the exponents equal to each other.
4. Solve for the variable: Isolate the variable to find its value.

Example: Solve for x in the equation: 4^(2x) = 16.

• Rewrite 16 as a power of 4: 16 = 4^2.
• The equation becomes: 4^(2x) = 4^2.
• Since the bases are the same, equate the exponents: 2x = 2.
• Solve for x: x = 1.

This method is effective for various equations involving indices, so practice with different bases to strengthen your understanding!

Indices, or exponents, are a way to express repeated multiplication. The laws of indices help simplify expressions involving powers. Here are the key laws:

• Product of Powers: When multiplying like bases, add the exponents.
  • Example: a^m × a^n = a^(m+n)
• Quotient of Powers: When dividing like bases, subtract the exponents.
  • Example: a^m ÷ a^n = a^(m-n)
• Power of a Power: When raising a power to another power, multiply the exponents.
  • Example: (a^m)^n = a^(m*n)
• Power of a Product: When raising a product to a power, distribute the exponent to each factor.
  • Example: (ab)^n = a^n * b^n
• Power of a Quotient: When raising a quotient to a power, distribute the exponent to the numerator and denominator.
  • Example: (a/b)^n = a^n / b^n

Using these laws can simplify complex expressions and make calculations easier.