Indices and Surds

Subject: Additional Mathematics
Topic: 4
Cambridge Code: 4037 / 0606

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Indices (Exponents)

Index/Exponent - Power to which a number is raised

Basic Notation

$a^n = a \times a \times a \times \cdots \times a \quad (n \text{ times})$

where $a$ is the base and $n$ is the index

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Laws of Indices

Law 1: Multiplication

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

Example: $2^3 \times 2^5 = 2^{3+5} = 2^8 = 256$

Law 2: Division

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

Example: $\frac{3^7}{3^4} = 3^{7-4} = 3^3 = 27$

Law 3: Power of a Power

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

Example: $(2^3)^2 = 2^{3 \times 2} = 2^6 = 64$

Law 4: Power of a Product

$(ab)^n = a^n b^n$

Example: $(2 \times 3)^2 = 2^2 \times 3^2 = 4 \times 9 = 36$

Law 5: Power of a Quotient

$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

Example: $\left(\frac{2}{3}\right)^2 = \frac{4}{9}$

Law 6: Zero Exponent

$a^0 = 1 \quad (a \neq 0)$

Example: $5^0 = 1$

Law 7: Negative Exponent

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

Example: $2^{-3} = \frac{1}{8}$

Law 8: Fractional Exponent

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

Example: $8^{\frac{2}{3}} = \sqrt[3]{8^2} = \sqrt[3]{64} = 4$

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Fractional Indices

Square Root

$a^{\frac{1}{2}} = \sqrt{a}$

Cube Root

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

General Form

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

Combined Form

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

Example Calculations

$16^{\frac{3}{4}} = \sqrt[4]{16^3} = \sqrt[4]{4096}$

Or alternatively: $16^{\frac{3}{4}} = (16^{\frac{1}{4}})^3 = 2^3 = 8$

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Surds (Radicals)

Surd - An irrational number containing a root that cannot be simplified to a rational number

Types of Surds

$\sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt[3]{2}, \sqrt[4]{3}, \text{ etc.}$

Note: $\sqrt{4} = 2$ is not a surd (it's rational)

Laws of Surds

Multiplication

$\sqrt{a} \times \sqrt{b} = \sqrt{ab}$

Example: $\sqrt{2} \times \sqrt{3} = \sqrt{6}$

Division

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

Example: $\frac{\sqrt{8}}{\sqrt{2}} = \sqrt{4} = 2$

Simplifying Surds

Separate perfect square factors from surds

Example 1: $\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$

Example 2: $\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$

Example 3: $\sqrt{50x^2} = \sqrt{25 \times 2 \times x^2} = 5x\sqrt{2}$

Adding and Subtracting Surds

Can only combine like surds

Example 1: $2\sqrt{3} + 5\sqrt{3} = 7\sqrt{3}$

Example 2: $\sqrt{8} + \sqrt{2} = 2\sqrt{2} + \sqrt{2} = 3\sqrt{2}$

Cannot combine: $\sqrt{2} + \sqrt{3}$ (different surds)

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Rationalizing Denominators

Rationalizing - Removing surds from denominator

Type 1: Single Surd in Denominator

Multiply by $\frac{\sqrt{a}}{\sqrt{a}}$

Example: $\frac{1}{\sqrt{2}} = \frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}$

Type 2: Sum or Difference with Surd

Use conjugate: multiply by $\frac{a \mp b}{a \mp b}$

Example: $\frac{2}{1+\sqrt{3}} = \frac{2}{1+\sqrt{3}} \times \frac{1-\sqrt{3}}{1-\sqrt{3}}$

$= \frac{2(1-\sqrt{3})}{1-3} = \frac{2(1-\sqrt{3})}{-2} = -(1-\sqrt{3}) = \sqrt{3}-1$

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Applications

Solving Equations with Indices

Example: Solve $2^x = 16$ $2^x = 2^4$ $x = 4$

Example: Solve $4^x = \frac{1}{8}$ $(2^2)^x = 2^{-3}$ $2^{2x} = 2^{-3}$ $2x = -3$ $x = -\frac{3}{2}$

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Key Points to Remember

1. Laws of indices are tools for simplification
2. Fractional indices relate to roots: $a^{\frac{1}{n}} = \sqrt[n]{a}$
3. Surds are irrational roots
4. Simplify surds by extracting perfect squares
5. Combine like surds only
6. Rationalize denominator where required

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Worked Examples

Example 1: Simplify

Simplify $\frac{a^6b^{-2}}{a^{-1}b^3}$

$= a^6 \times a^{-(-1)} \times b^{-2} \times b^{-3}$ $= a^{6+1} \times b^{-2-3}$ $= a^7b^{-5} = \frac{a^7}{b^5}$

Example 2: Fractional Indices

Solve $x^{\frac{2}{3}} = 4$

Taking both sides to power $\frac{3}{2}$: $(x^{\frac{2}{3}})^{\frac{3}{2}} = 4^{\frac{3}{2}}$ $x = 4^{\frac{3}{2}} = (4^{\frac{1}{2}})^3 = 2^3 = 8$

Example 3: Rationalize

Rationalize $\frac{3}{2\sqrt{5}}$

$= \frac{3}{2\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{3\sqrt{5}}{10}$

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Practice Questions

1. Simplify:

   • $(2^4)^2 \times 2^{-3}$
   • $\frac{x^5y^{-2}}{x^{-1}y^4}$

2. Simplify surds:

   • $\sqrt{24}$
   • $\sqrt{48x^3}$

3. Rationalize:

   • $\frac{5}{\sqrt{2}}$
   • $\frac{1}{3-\sqrt{2}}$

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Revision Tips

• Remember all 8 laws of indices
• Fractional indices = roots
• Simplify surds by factoring
• Rationalizing: multiply by conjugate if needed
• Practice index equations