Index Notation and Powers of 10

(Note: Index, Power or Exponent all mean the same thing)

The index of a number shows you how many times to use
the number in a multiplication.

This means 10 × 10 (10 is used 2 times in the multiplication)

Example 1: 10³ = 10 × 10 × 10 = 1,000

• In words: 10³ could be called "10 to the third power", "10 to the power 3" or simply "10 cubed"

Example 2: 10⁴ = 10 × 10 × 10 × 10 = 10,000

• In words: 10⁴ could be called "10 to the fourth power", "10 to the power 4" or simply "10 to the 4"

You can multiply any number by itself as many times as you want using this notation (see Exponents), but powers of 10 have a special use ...

Powers of 10

"Powers of 10" is a very useful way of writing down large numbers.

Instead of having lots of zeros, you show how many powers of 10 you need to make that many zeros

Example: 5,000 = 5 × 1,000 = 5 × 10³

• 5 thousand is 5 times a thousand. And a thousand is 10³. So 5 times 10³ = 5,000
• Can you see that 10³ is a handy way of making 3 zeros?

Scientists and Engineers (who often use very big or very small numbers) find it very useful to write numbers this way, such as:

• 9.46 x 10¹⁵ meters (the distance light travels in one year), or
• 1.9891 x 10³⁰ kg (the mass of the Sun).

It saves them writing down lots of zeros. It is commonly called Scientific Notation, or Standard Form.

While at first it may look hard, there is an easy "trick":

Example: What is 1.35 × 10⁴ ?

You can calculate it as: 1.35 x (10 × 10 × 10 × 10) = 1.35 x 10,000 = 13,500

But it is easier to think "move the decimal point 4 places to the right" like this:

1.35

13.5

135.

1350.

13500.

Negative Powers of 10

Negative? What could be the opposite of multiplying? Dividing!

A negative power means how many times to divide by the number.

Negatives just go the other way!

Example: 5 × 10^-3 = 5 ÷ 10 ÷ 10 ÷ 10 = 0.005

Just remember for negative powers of 10:

For negative powers of 10, move the decimal point to the left.

Example: What is 7.1 × 10^-3?

Well, it is really 7.1 x (¹/₁₀ × ¹/₁₀ × ¹/₁₀) = 7.1 x 0.001 = 0.0071

But it is easier to think "move the decimal point 3 places to the left" like this:

7.1

0.71

0.071

0.0071

Animation

Summary