# Surds Indices

## Surds Indices

Understanding Surds and Indices

• Surds: Are irrational numbers that, when in decimal form, are non-repeating and non-terminating. They are often the square root (or other roots) of a non-square integer.
• Indices: Powerful tools used in algebra to express repeated multiplication, often known as powers or exponents.

Types of Surds

• Perfect Surds: are the square roots of perfect square numbers. For example, √16 = 4 is a perfect surd.
• Imperfect Surds: are the square roots of a number that is not a perfect square. Such as, √2.

Properties of Indices

• Multiplication of indices: When you multiply two powers with the same base, you add the indices. For instance, a^m * a^n = a^(m+n).
• Division of indices: When you divide two powers with the same base, you subtract the indices. For instance, a^m / a^n = a^(m-n).
• Power of power rule: When you raise a power to another power, you multiply the indices. For example, (a^m)^n = a^(m*n).
• Zero exponent rule: any non-zero number raised to the power of zero is 1.

Simplifying and Rationalizing Surds

• Rationalizing the denominator: This process involves rewriting a surd so that the denominator is rational. It can be achieved by multiplying top and bottom by the conjugate of the denominator.
• Simplifying surds: This involves making the number inside the square root as small as possible. This process often includes factorisation.

Expressions Involving Surds and Indices

• Applying index laws: The index laws still apply when dealing with surds. The square root of a product can be written as the product of the square roots and vice versa.
• Solving equations with surds: Equations with surds can be solved by squaring both sides or by isolating the surd on one side of the equation and then squaring.