Mathematics (0580) | Indices I

Hello future mathematicians! Welcome to the exciting world of Indices (also called Exponents or Powers). This chapter is fundamental—it gives you a powerful tool to handle very large or very small numbers easily, a skill you’ll use throughout your IGCSE journey, especially in Algebra and Number work.

Think of indices as a mathematical shorthand. Instead of writing out long multiplications, we compress the information. Mastering the rules of indices will make complex calculations much simpler! Let’s get started.

Section 1: The Anatomy of an Index

1.1 What is an Index?

When we multiply a number by itself repeatedly, we use an index (or power) to show how many times the multiplication occurs.

Consider the expression: \(5^3\)

• The Base is 5. This is the number being multiplied.
• The Index (or Exponent or Power) is 3. This tells you how many times the base is multiplied by itself.

\(5^3 = 5 \times 5 \times 5 = 125\)
(We read this as "5 to the power of 3" or "5 cubed".)

Analogy: The Stacking Analogy
If you have a base (a floor) and you stack blocks on it, the index tells you the height of the stack. A stack of 4 blocks is \(b^4\).

1.2 Powers and Roots (Review)

Before diving into the rules, it’s useful to remember common powers (squares and cubes) and their corresponding roots.

• Squares: \(4^2 = 16\). The Square Root is the reverse: \(\sqrt{16} = 4\).
• Cubes: \(4^3 = 64\). The Cube Root is the reverse: \(\sqrt[3]{64} = 4\).
• The syllabus requires you to recall common squares (1 to 15) and cubes (1 to 10). (Example: Write down the value of \(\sqrt{169}\). Answer: 13, since \(13^2 = 169\).)

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Section 2: The Special Powers (Zero and Negative Indices)

These two types of indices are where the laws of mathematics get interesting and often confuse students. Don’t worry; they follow simple, non-negotiable rules!

2.1 The Zero Index Rule

This is perhaps the easiest rule in indices:

Any non-zero base raised to the power of zero is 1.

\[ a^0 = 1 \]
(where \(a \neq 0\))

Why is this true? (The Logic Chain)
Imagine the powers of 2:

• \(2^3 = 8\)
• \(2^2 = 4\) (We divide 8 by 2)
• \(2^1 = 2\) (We divide 4 by 2)
• \(2^0 = 1\) (We must divide 2 by 2)

It doesn't matter how complicated the base is:

• \(7^0 = 1\)
• \((15x)^0 = 1\)
• \((-450)^0 = 1\)

2.2 The Negative Index Rule

A negative index does not mean the answer is negative. It means you take the reciprocal (flip the fraction) of the base raised to the corresponding positive power.

\[ a^{-n} = \frac{1}{a^n} \]

Step-by-Step Process for Negative Indices:

1. Move the powered term to the opposite side of the fraction (e.g., from the numerator to the denominator).
2. Change the sign of the index from negative to positive.

Example 1: Find the value of \(5^{-2}\)
\(5^{-2} = \frac{1}{5^2} = \frac{1}{25}\)

Example 2: Simplify \(\frac{1}{x^{-3}}\)
\(\frac{1}{x^{-3}} = x^3\)

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Section 3: The Three Key Rules of Indices

These are the three fundamental laws you must memorize. They work for positive, negative, and zero indices.

3.1 Rule 1: Multiplication (The Addition Rule)

When multiplying terms with the same base, you add the indices.

\[ a^m \times a^n = a^{m+n} \]

Example: Simplify \(x^4 \times x^3\)
\(x^4 \times x^3 = x^{4+3} = x^7\)

Example with negative indices: Simplify \(3^{-2} \times 3^5\)
\(3^{-2} \times 3^5 = 3^{-2+5} = 3^3 = 27\)

3.2 Rule 2: Division (The Subtraction Rule)

When dividing terms with the same base, you subtract the indices.

\[ a^m \div a^n = a^{m-n} \]

Example 1: Simplify \(y^8 \div y^2\)
\(y^8 \div y^2 = y^{8-2} = y^6\)

Example 2: Simplify \(2^3 \div 2^5\)
\(2^3 \div 2^5 = 2^{3-5} = 2^{-2} = \frac{1}{2^2} = \frac{1}{4}\)

3.3 Rule 3: Power of a Power (The Multiplication Rule)

When raising a power to another power, you multiply the indices.

\[ (a^m)^n = a^{mn} \]

This rule also applies when a product is raised to a power:

\[ (ab)^n = a^n b^n \]
\[ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \]

Example 1: Simplify \((z^3)^4\)
\((z^3)^4 = z^{3 \times 4} = z^{12}\)

Example 2: Simplify \((2x^4)^3\)
\((2x^4)^3 = 2^3 \times (x^4)^3 = 8x^{12}\)

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Section 4: Fractional Indices (Extended Content Only)

If you are taking the Core paper, you only need to know positive, negative, and zero indices (Sections 1, 2, and 3). If you are taking the Extended paper, you need this section!

4.1 The Denominator Rule: Roots

A fractional index represents a root. Specifically, the denominator of the fraction tells you which root to take.

\[ a^{1/n} = \sqrt[n]{a} \]

Example 1: Find the value of \(9^{1/2}\)
The denominator is 2, so we take the square root.
\(9^{1/2} = \sqrt[2]{9} = 3\)

Example 2: Find the value of \(64^{1/3}\)
The denominator is 3, so we take the cube root.
\(64^{1/3} = \sqrt[3]{64} = 4\)

Did you know? The notation $\sqrt{a}$ is actually shorthand for $a^{1/2}$!

4.2 Combining Power and Root

When you have a fractional index with a numerator other than 1, like $m/n$, it means you must apply both a root (using the denominator $n$) and a power (using the numerator $m$).

\[ a^{m/n} = (\sqrt[n]{a})^m \]

Rule of Thumb: Do the Root First!
It is almost always easier to calculate the root first to get a smaller number, and then apply the power.

Example: Work out \(8^{2/3}\)

Step 1: Identify the root (denominator).
The denominator is 3, so we take the cube root of 8.
\(\sqrt[3]{8} = 2\)

Step 2: Apply the power (numerator).
The numerator is 2, so we square the result.
\(2^2 = 4\)

Therefore, \(8^{2/3} = 4\).

4.3 Fractional and Negative Indices Together

If the index is negative AND fractional, apply the negative index rule (reciprocal/flip) first, then deal with the fraction.

\[ a^{-m/n} = \frac{1}{a^{m/n}} = \frac{1}{(\sqrt[n]{a})^m} \]

Example: Work out \(16^{-3/4}\)

Step 1: Deal with the negative sign (Reciprocal).
\(16^{-3/4} = \frac{1}{16^{3/4}}\)

Step 2: Deal with the root (Denominator is 4).
\(\sqrt[4]{16} = 2\) (Since \(2 \times 2 \times 2 \times 2 = 16\))

Step 3: Deal with the power (Numerator is 3).
\(2^3 = 8\)

Final Answer: \(\frac{1}{8}\)

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