Welcome to the World of Indices and Standard Form!

Ever wondered how scientists talk about the distance to the sun or the size of a tiny atom without writing out dozens of zeros? They use indices and standard form. In this chapter, we will learn how to handle very large and very small numbers with ease. These skills are essential for both Paper 1 and Paper 2 of your Edexcel IGCSE Mathematics (Specification B) exam.

Note: For more on surds (like \(\sqrt{3}\)) and how to simplify them, please see the separate chapter on "Surds and rationalising the denominator".

1. Understanding Indices (Powers)

An index (plural: indices), also known as a power or exponent, tells us how many times a number is multiplied by itself. In the expression \(a^n\):

  • \(a\) is the base (the number being multiplied).
  • \(n\) is the index (the number of times it appears).

Example: \(5^3 = 5 \times 5 \times 5 = 125\)

The Fundamental Laws of Indices

When working with indices, there are three main rules you must remember. These work as long as the base is the same.

Rule 1: Multiplication
When multiplying, add the indices.
\(a^m \times a^n = a^{m+n}\)
Example: \(2^3 \times 2^4 = 2^{3+4} = 2^7\)

Rule 2: Division
When dividing, subtract the indices.
\(a^m \div a^n = a^{m-n}\)
Example: \(5^6 \div 5^2 = 5^{6-2} = 5^4\)

Rule 3: Power of a Power
When raising a power to another power, multiply the indices.
\((a^m)^n = a^{m \times n}\)
Example: \((3^2)^4 = 3^{2 \times 4} = 3^8\)

Quick Takeaway: Always check that the bases are identical before applying these rules! You can't combine \(2^3 \times 3^2\) using these laws because the bases (2 and 3) are different.

2. Special Indices: Zero, Negative, and Fractions

Don't let "weird-looking" powers scare you. They follow very logical patterns.

The Zero Index

Any non-zero number raised to the power of zero is 1.
\(a^0 = 1\)
Example: \(100^0 = 1\) and \((\frac{1}{2})^0 = 1\)

Negative Indices

A negative index represents a reciprocal (one over the number). It does not make the number negative!
\(a^{-n} = \frac{1}{a^n}\)
Example: \(4^{-2} = \frac{1}{4^2} = \frac{1}{16}\)

Fractional Indices (Roots)

Fractional indices are just another way of writing roots.
\(a^{1/n} = \sqrt[n]{a}\)
Example: \(9^{1/2} = \sqrt{9} = 3\)
Example: \(8^{1/3} = \sqrt[3]{8} = 2\)

Combined Fractional Indices:
If the fraction has a top number (numerator) other than 1, use this rule: "Root the bottom, power the top."
\(a^{m/n} = (\sqrt[n]{a})^m\)
Example: \(25^{3/2} = (\sqrt{25})^3 = 5^3 = 125\)

Common Mistake to Avoid: Students often think \(5^{-2}\) is \(-25\). Remember: a negative power "flips" the number into a fraction; it doesn't change the sign of the base!

3. Standard Form

Standard Form (also called scientific notation) is a way of writing very large or very small numbers neatly. A number in standard form always looks like this:

\(a \times 10^n\)

There are two strict rules for this format:

  1. The value of \(a\): It must be at least 1 but less than 10 (\(1 \le a < 10\)).
  2. The value of \(n\): It must be an integer (a whole number).

Converting to Standard Form

For Large Numbers: Count how many places you move the decimal point to the left until only one digit remains before the decimal.
Example: \(45,000 = 4.5 \times 10^4\)

For Small Numbers: Count how many places you move the decimal point to the right until it is just after the first non-zero digit. The power will be negative.
Example: \(0.00072 = 7.2 \times 10^{-4}\)

Calculations with Standard Form

You can use your index laws to solve these quickly, especially in Paper 1.

Multiplication: Multiply the numbers, add the powers of 10.
\((2 \times 10^3) \times (3 \times 10^4) = (2 \times 3) \times 10^{3+4} = 6 \times 10^7\)

Division: Divide the numbers, subtract the powers of 10.
\((8 \times 10^6) \div (2 \times 10^2) = (8 \div 2) \times 10^{6-2} = 4 \times 10^4\)

Top Tip: After calculating, always check if your answer is still in standard form! If you get \(12 \times 10^5\), you must change it to \(1.2 \times 10^6\) to be correct.

4. Quick Review Box

Key Terms:
  • Base: The main number.
  • Index/Power: The small number at the top.
  • Reciprocal: \(1\) divided by the number (linked to negative powers).
  • Standard Form: \(a \times 10^n\) where \(1 \le a < 10\).
Summary of Laws:
  • Multiply \(\implies\) Add powers
  • Divide \(\implies\) Subtract powers
  • Power of a power \(\implies\) Multiply powers
  • Negative power \(\implies\) Fraction (\(1\) over ...)
  • Fractional power \(\implies\) Root

Did you know? Standard form is used by computer scientists to represent "Floating Point" numbers, which allows computers to handle massive calculations without crashing!

Check your understanding: Can you explain why \(7^{-1}\) is the same as \(\frac{1}{7}\)? If you can, you've mastered the logic of indices!