Powers and Roots

What Are Powers?

When you multiply a number by itself repeatedly, you can write it using powers (also called indices or exponents).

For example:

• 2 × 2 × 2 × 2 = 2⁴ = 16

In the expression 2⁴:

• 2 is called the base (the number being multiplied by itself)
• 4 is called the index or power (it tells you how many times to multiply the base by itself)

We read 2⁴ as "2 to the power of 4" or "2 raised to the power 4".

Important facts:

• Any number to the power of 1 equals itself: 6¹ = 6
• Any number (except zero) to the power of 0 equals 1: 3⁰ = 1

Square Numbers and Square Roots

What is a Square Root?

A square root is the opposite of squaring. It's the number that was multiplied by itself to give you the square number.

For example:

• The square root of 25 is 5, because 5 × 5 = 25

The symbol for square root is √.

So: √25 = 5

Important points about square roots:

• Every positive number actually has TWO square roots: one positive and one negative

• For example, both 5 and -5 are square roots of 25, because:

  • 5 × 5 = 25
  • (-5) × (-5) = 25

• However, when you see the √ symbol, it ALWAYS means the positive square root only

• So √25 = 5 (not -5)

• If you want to show both roots, use the ± symbol (read as "plus or minus")

• The square roots of 25 are ±5, meaning 5 and -5

Squares and Square Roots You Must Know

You must memorize these squares and their square roots (from 1 to 15):

Exam Tip: Practice these until you can recall them instantly. You will need them for both calculator and non-calculator papers.