Laws of Indices and Rules for Surds

Flashcards covering key index laws, expanding expressions with fractional exponents, and rules for manipulating and simplifying surds.

$$a^{-n}$$

The negative index rule which states that a base raised to a negative power equals $\frac{1}{a^n}$.

$$a^m \times a^n$$

The multiplication rule for indices which simplifies by adding the exponents to yield $a^{m+n}$.

$$a^m \div a^n$$

The division rule for indices which simplifies by subtracting the exponents to yield $a^{m-n}$.

$$a^{\frac{m}{n}}$$

The fractional index rule where the denominator indicates the root and the numerator indicates the power, written as $\sqrt[n]{a^m}$.

Power Outside of Brackets Rule with Addition

A rule stating that you cannot apply the power outside of the brackets directly to all individual terms inside the brackets if there is an addition sign separating them.

Expansion of $(3x^{\frac{3}{2}} - x^{-\frac{1}{2}})^2$

An algebraic method where the bracket is multiplied by itself as $(3x^{\frac{3}{2}} - x^{-\frac{1}{2}})(3x^{\frac{3}{2}} - x^{-\frac{1}{2}})$ rather than distributing the outer exponent to each term directly.

$$\sqrt{ab}$$

The surd multiplication rule stating that the square root of a product can be split into $\sqrt{a}\sqrt{b}$.

$$\sqrt{a} \times \sqrt{a}$$

The surd rule stating that multiplying the square root of a number by itself equals the original number, $a$.

$$\sqrt{\frac{a}{b}}$$

The surd quotient rule stating that the square root of a fraction equals $\frac{\sqrt{a}}{\sqrt{b}}$.

$$b\sqrt{a} + c\sqrt{a}$$

The rule for adding like surds by combining their coefficients, resulting in $(b+c)\sqrt{a}$.

Rationalising the Denominator

A surd operation alongside simplifying surds used to eliminate radical expressions from the denominator of a fraction.