Introduction to Powers, Roots, and Indices

Welcome! In this chapter, we are looking at Powers, Roots, and Indices. Think of these as mathematical "shortcuts." Instead of writing out a long string of numbers to multiply, we use powers to keep things neat and tidy. Whether you are aiming for a Grade 1 or a Grade 9, mastering these basics is essential because they pop up in almost every other area of Maths, from Algebra to Standard Form.

Note: If you need a refresher on basic multiplication, check out the chapter on "Integers, decimals and fractions."

1. Squares, Cubes, and Beyond

A power (also called an index or exponent) tells you how many times to multiply a number by itself. The large number is called the base, and the small lifted number is the index.

Example: In \(5^3\), the base is \(5\) and the index is \(3\). It means \(5 \times 5 \times 5 = 125\).

Square Numbers (up to \(15 \times 15\))

For your GCSE, you are expected to know your square numbers up to \(15^2\) by heart. These are "standard" content for all students:

  • \(1^2 = 1\)
  • \(2^2 = 4\)
  • \(3^2 = 9\)
  • \(4^2 = 16\)
  • \(5^2 = 25\)
  • \(6^2 = 36\)
  • \(7^2 = 49\)
  • \(8^2 = 64\)
  • \(9^2 = 81\)
  • \(10^2 = 100\)
  • \(11^2 = 121\)
  • \(12^2 = 144\)
  • \(13^2 = 169\)
  • \(14^2 = 196\)
  • \(15^2 = 225\)

Cube Numbers and Others

You also need to recognise powers of \(2, 3, 4,\) and \(5\). Specifically, you should know cube numbers (\(base^3\)) up to \(5^3\):

  • \(1^3 = 1\)
  • \(2^3 = 8\)
  • \(3^3 = 27\)
  • \(4^3 = 64\)
  • \(5^3 = 125\)

Quick Review: A power of \(1\) doesn't change the number: \(7^1 = 7\). Any number (except zero) to the power of \(0\) is always \(1\): \(5^0 = 1\).

2. Understanding Roots

A root is the "inverse" (the opposite) of a power. If you know that \(4^2 = 16\), then the square root of \(16\) is \(4\).

Notation: We use the symbol \(\sqrt{}\) for square roots and \(\sqrt[3]{}\) for cube roots.

  • \(\sqrt{81} = 9\) (because \(9 \times 9 = 81\))
  • \(\sqrt[3]{27} = 3\) (because \(3 \times 3 \times 3 = 27\))

Did you know? Positive numbers actually have two square roots: a positive one and a negative one. For example, \(\sqrt{25}\) could be \(5\) or \(-5\), because \(-5 \times -5\) also equals \(25\). In your exam, usually the positive root is expected unless the question asks for both!

3. The Laws of Indices

When we multiply or divide numbers with the same base, we can use these three golden rules. These are vital for both the non-calculator and calculator papers.

Rule 1: Multiplication (Add the powers)

When multiplying, if the bases are the same, add the indices: \(a^m \times a^n = a^{m+n}\)

Example: \(3^2 \times 3^4 = 3^{(2+4)} = 3^6\)

Rule 2: Division (Subtract the powers)

When dividing, if the bases are the same, subtract the indices: \(a^m \div a^n = a^{m-n}\)

Example: \(5^7 \div 5^3 = 5^{(7-3)} = 5^4\)

Rule 3: Brackets (Multiply the powers)

When a power is outside a bracket, multiply the indices: \((a^m)^n = a^{m \times n}\)

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

Common Mistake to Avoid: These rules only work if the base is the same. You cannot simplify \(2^3 \times 5^2\) using these laws!

4. Negative and Fractional Indices (Higher Tier Only)

If you are taking the Higher Tier, you need to handle indices that aren't just positive whole numbers.

Negative Indices

A negative index tells you to find the reciprocal (flip the number into a fraction). It does not make the number negative!

\(a^{-n} = \frac{1}{a^n}\)

Example: \(5^{-2} = \frac{1}{5^2} = \frac{1}{25}\)

Fractional Indices

Fractional indices represent roots. The number on the bottom (the denominator) is the "root," and the number on the top (the numerator) is the "power."

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

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

Example: \(16^{\frac{1}{2}} = \sqrt{16} = 4\)

Example: \(8^{\frac{2}{3}}\). First, cube root the \(8\) to get \(2\). Then, square that answer: \(2^2 = 4\). So, \(8^{\frac{2}{3}} = 4\).

Memory Aid: For fractional indices, think of a tree. The roots are at the bottom, and the leaves (power) are at the top!

5. Estimating Powers and Roots (Higher Tier Only)

Sometimes you won't have a calculator and you'll be asked to estimate a root that isn't a perfect square.

Step-by-step: How to estimate \(\sqrt{50}\)

  1. Find the two perfect squares that \(50\) sits between. In this case, \(49\) (\(7^2\)) and \(64\) (\(8^2\)).
  2. Since \(50\) is very close to \(49\), \(\sqrt{50}\) must be just slightly larger than \(7\).
  3. A good estimate would be \(7.1\).

Summary Checklist

  • Basics: Do you know your squares to \(15 \times 15\) and cubes to \(5^3\)?
  • Laws: Do you remember to add when multiplying and subtract when dividing?
  • Zero: Remember that any number to the power of \(0\) is \(1\).
  • (Higher Only) Negative: Does a negative index flip the number into a fraction?
  • (Higher Only) Fractions: Is the bottom number the root and the top number the power?

Key Takeaway: Powers and roots are just different ways of looking at multiplication. Practice the "Laws of Indices" until they become second nature, and always double-check if a power is negative before you start calculating!