Laws of Indices and Surds

Laws of Indices

Any number to the power of 0 is 1. For example: 4^0 = 1.
Multiplying powers with the same base: Add the exponents. This can be written as a^m × a^n = a^(m+n).
Dividing powers with the same base: Subtract the exponents. This can be written as a^m ÷ a^n = a^(m-n).
Raising a power to a power: Multiply the exponents. This can be written as (a^m)^n = a^(mn).
Negative exponent rule: A number with a negative exponent can be written as 1 divided by the number with the positive of that exponent. This can be written as a^-m = 1 / a^m.

Squaring a surd: If you square a square root, you get the value under the root. In other words, (√a)^2 = a.
Multiplication of surds: The product of the square roots of two numbers is the square root of their product. This can be written as √a × √b = √(ab).
Division of surds: The quotient of the square roots of two numbers is the square root of their quotient. This can be written as √a ÷ √b = √(a/b).
Rationalising the denominator: Conversion of a denominator that contains a surd to one that no longer contains a surd.

All the laws of indices and surds are interlinked, and understanding their relationships will benefit problem-solving and exam performance.
Ensure you can apply them systematically and calculate accurately when using them, particularly when a question combines multiple laws.