Surds and Indices

What are Indices (Exponents)?

Indices (also called powers or exponents) tell you how many times to multiply a number by itself.

• $a^n$ means $a \times a \times a \ldots \times a$ (n times)

Example:

$$2^3 = 2 \times 2 \times 2 = 8$$

What are Surds?

Surds are irrational numbers that cannot be simplified to remove the square root (or cube root, etc.) and still remain rational.

• $\sqrt{2}, \sqrt{3}, \sqrt{5}$ are surds
• $\sqrt{4} = 2$ is not a surd (it’s rational)

Key Insight: Surds are used when exact decimal values aren't possible or useful. They maintain precision.

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Key Formulas and Laws

Laws of Indices (Exponents)

$$a^m \times a^n = a^{m+n} \\ \frac{a^m}{a^n} = a^{m-n} \\ (a^m)^n = a^{mn} \\ (ab)^n = a^n \cdot b^n \\ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$

Negative and Fractional Indices

$$a^{-n} = \frac{1}{a^n} \\ a^{0} = 1 \\ a^{\frac{1}{n}} = \sqrt[n]{a} \\ a^{\frac{m}{n}} = \sqrt[n]{a^m}$$

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Surds: Key Concepts

Simplest Form of Surds

• $\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$
• Always factor out perfect squares from under the root

Addition/Subtraction of Surds

Only like surds can be added or subtracted (same irrational part).

$$3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2} \\ 4\sqrt{3} - 2\sqrt{3} = 2\sqrt{3}$$

Multiplication of Surds

$$\sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \quad (\text{valid only when } a, b \geq 0)$$

Division of Surds

$$\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \quad (\text{rationalize if needed})$$

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Rationalization

Rationalizing the denominator means eliminating surds from the denominator.

Cases:

1. Single term in denominator:

$$\frac{1}{\sqrt{2}} = \frac{1 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{\sqrt{2}}{2}$$

2. Binomial surd in denominator (conjugate method):

$$\frac{1}{\sqrt{2} + 1} = \frac{1(\sqrt{2} - 1)}{(\sqrt{2} + 1)(\sqrt{2} - 1)} = \frac{\sqrt{2} - 1}{2 - 1} = \sqrt{2} - 1$$

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Visual Understanding

• $\sqrt{2}$ is approximately 1.414, an irrational number that continues infinitely
• On the number line, irrational numbers lie between rational numbers but cannot be expressed as exact fractions
• Indices compress long multiplication; surds preserve exactness when roots can’t be simplified

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Conceptual Traps to Avoid

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Examples

Example 1: Simplify $\sqrt{75} + \sqrt{12}$

$$\sqrt{75} = \sqrt{25 \cdot 3} = 5\sqrt{3} \\ \sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3} \\ \Rightarrow 5\sqrt{3} + 2\sqrt{3} = 7\sqrt{3}$$

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Example 2: Simplify $(3\sqrt{2})(2\sqrt{3})$

$$3 \cdot 2 \cdot \sqrt{2} \cdot \sqrt{3} = 6\sqrt{6}$$

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Example 3: Rationalize $\frac{1}{3 + \sqrt{2}}$

Multiply numerator and denominator by conjugate:

$$\frac{1 \cdot (3 - \sqrt{2})}{(3 + \sqrt{2})(3 - \sqrt{2})} = \frac{3 - \sqrt{2}}{9 - 2} = \frac{3 - \sqrt{2}}{7}$$

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Example 4: Simplify $16^{3/4}$

$$16 = 2^4 \Rightarrow 16^{3/4} = (2^4)^{3/4} = 2^3 = 8$$

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Example 5: Convert to root form: $27^{2/3}$

$$27^{2/3} = \sqrt[3]{27^2} = \sqrt[3]{729} = 9$$