Laws of Indices and Surds

A comprehensive set of flashcards covering the key rules and worked examples for indices and surds, including multiplication and division of powers, fractional and negative indices, simplification of surds, and rationalising denominators.

What is the rule for multiplying powers with the same base?

a^m × a^n = a^(m+n) (add the powers).

What is the rule for dividing powers with the same base?

a^m ÷ a^n = a^(m-n) (subtract the powers).

What is the rule for raising a power to another power?

(a^m)^n = a^(mn) (multiply the powers).

What does a negative index signify?

a^(-n) = 1 / a^n.

How is a fractional index interpreted?

a^(m/n) = (n-th root of a)^m.

What is any non-zero number raised to the power of zero?

a^0 = 1 (provided a ≠ 0).

What is the product rule for square roots (surds)?

√(ab) = √a × √b.

What is the quotient rule for square roots?

√(a/b) = √a ÷ √b.

How do you simplify √12?

√12 = √(4 × 3) = 2√3.

How do you multiply surds such as √3 × √6?

√3 × √6 = √(3 × 6) = √18 = 3√2.

What does it mean to "rationalise the denominator"?

To remove any surd from the denominator of a fraction.

How do you rationalise 9 / √3?

Multiply by √3/√3: (9√3)/3 = 3√3.

How do you rationalise 1 / (1 − √2)?

Multiply top and bottom by (1 + √2): result = −1 − √2.

Simplify x³ × x⁵.

x³ × x⁵ = x^(3+5) = x⁸.

Simplify √28.

√28 = √(4 × 7) = 2√7.

Simplify 6√3 + 2√27.

2√27 = 6√3, so 6√3 + 6√3 = 12√3.

Rationalise 2 / (3 + √7).

Multiply by (3 − √7)/(3 − √7): (2(3 − √7))/(9 − 7) = (6 − 2√7)/2 = 3 − √7.