Algebraic Root & Indices

Laws of Indices

Indices (or powers) show how many times a number or algebraic expression is multiplied by itself. The laws of indices allow us to simplify expressions involving powers.

Product Rule

When multiplying two powers with the same base, add the indices:

$$a^m \times a^n = a^{m+n}$$

For example, $x^3 \times x^2 = x^{3+2} = x^5$.

Quotient Rule

When dividing two powers with the same base, subtract the indices:

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

For example, $\frac{y^5}{y^2} = y^{5-2} = y^3$.

Power of a Power

When raising a power to another power, multiply the indices:

$$(a^m)^n = a^{mn}$$

For example, $(x^2)^3 = x^{2 \times 3} = x^6$.

Zero and Negative Indices

Zero index: Any non-zero base raised to the zero power equals 1:

$$a^0 = 1 \quad \text{(where } a \neq 0\text{)}$$

For example, $5^0 = 1$.

Negative index: A negative index means the reciprocal of the positive index:

$$a^{-n} = \frac{1}{a^n}$$

For example, $x^{-3} = \frac{1}{x^3}$.

Simplifying Expressions with Indices

Roots can be expressed using fractional indices, which helps simplify powers and roots together.

Expressing Roots as Fractional Indices

The square root and cube root can be written as powers with fractional indices:

• $\sqrt{a} = a^{\frac{1}{2}}$
• $\sqrt[3]{a} = a^{\frac{1}{3}}$

More generally, the nth root of $a$ is:

$$\sqrt[n]{a} = a^{\frac{1}{n}}$$

Simplifying Powers and Roots

You can combine powers and roots using the laws of indices. For example:

$$a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}$$

This means you can rewrite powers with fractional indices as roots raised to powers, or vice versa.

For instance, $16^{\frac{3}{4}}$ can be written as $\left(\sqrt[4]{16}\right)^3$. Since $\sqrt[4]{16} = 2$, then $16^{\frac{3}{4}} = 2^3 = 8$.

Combining Like Terms with Indices

When terms have the same base and index, you can combine them by adding or subtracting their coefficients:

For example, $3x^2 + 5x^2 = 8x^2$.

However, you cannot combine terms with different indices or bases directly, e.g. $x^2 + x^3$ stays as it is.

For example, $2a^3 + 4a^3 = 6a^3$.

For example, simplifying $9^{\frac{1}{2}} \times 9^{\frac{3}{2}}$:

$$9^{\frac{1}{2}} \times 9^{\frac{3}{2}} = 9^{\frac{1}{2} + \frac{3}{2}} = 9^2 = 81$$

Operations with Roots

Roots (square roots, cube roots, etc.) can be simplified and manipulated using surds and rationalisation techniques.

A surd is an irrational root that cannot be simplified to a rational number.

Square Roots and Cube Roots

The square root of a number $a$ is a value $b$ such that $b^2 = a$. Similarly, the cube root of $a$ is a value $c$ such that $c^3 = a$.

For example, $\sqrt{25} = 5$ because $5^2 = 25$, and $\sqrt[3]{27} = 3$ because $3^3 = 27$.

Simplifying Surds

To simplify surds:

• Factor the number inside the root into its prime factors.
• Pair factors for square roots (or group in threes for cube roots) to take them outside the root.

For example, simplify $\sqrt{72}$:

$$72 = 36 \times 2$$

$$\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}$$

Rationalising Denominators

Rationalising the denominator means rewriting a fraction so that the denominator contains no surds.

If the denominator is a single surd, multiply numerator and denominator by that surd:

$$\frac{1}{\sqrt{a}} \times \frac{\sqrt{a}}{\sqrt{a}} = \frac{\sqrt{a}}{a}$$

If the denominator is a binomial with surds, multiply numerator and denominator by the conjugate (change the sign between terms):

$$\frac{1}{a + \sqrt{b}} \times \frac{a - \sqrt{b}}{a - \sqrt{b}} = \frac{a - \sqrt{b}}{a^2 - b}$$

For example, rationalise $\frac{5}{\sqrt{3}}$:

$$\frac{5}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{3}$$