# Index Notation and Powers of 10

The exponent (or index or power) of a number says**how many times** to use the number in a **multiplication**.

10^{2} means **10** × **10** = 100

(It says **10** is used **2** times in the multiplication)

### Example: 10^{3} = 10 × 10 × 10 = 1,000

- In words: 10
^{3}could be called "10 to the third power", "10 to the power 3" or simply "10 cubed"

### Example: 10^{4} = 10 × 10 × 10 × 10 = 10,000

- In words: 10
^{4}could be called "10 to the fourth power", "10 to the power 4" or "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 or small numbers.

Instead of having lots of zeros, you show how many **powers of 10** will make that many zeros

### Example: 5,000 = 5 × 1,000 = 5 × 10^{3}

5 thousand is 5 times a thousand. And a thousand is 10^{3}. So 5 times 10^{3} = 5,000

Can you see that 10^{3} is a handy way of making 3 zeros?

Scientists and Engineers (who often use very big or very small numbers) like to write numbers this way.

### Example: The Mass of the Sun

The Sun has a Mass of 1.988 × 10^{30} kg.

It is too hard to write 1,988,000,000,000,000,000,000,000,000,000 kg

(And very easy to make a mistake counting the zeros!)

### Example: A Light Year (the distance light travels in one year)

It is easier to use **9.461 × 10 ^{15}** meters, rather than

**9,461,000,000,000,000**meters

It is commonly called **Scientific Notation**, or **Standard Form.**

## Other Way of Writing It

Sometimes people use the **^** symbol (above the 6 on your keyboard), as it is easy to type.

### Example: **3 × ****10^4 is the same as 3 × 10**^{4}

^{4}

**3 × 10^4 = 3 × 10 × 10 × 10 × 10 = 30,000**

Calculators often use "E" or "e" like this:

### Example: **6E+****5 is the same as 6 × 10**^{5}

^{5}

**6E+5 = 6 × 10 × 10 × 10 × 10 × 10 = 600,000**

### Example: **3.12E4**** is the same as 3.12 × 10**^{4}

^{4}

**3.12E4 = 3.12 × 10 × 10 × 10 × 10 = 31,200**

## The Trick

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

The index of 10 says ...

... **how many places** to move the decimal point to the right.

### Example: What is 1.35 × 10^{4} ?

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.

### 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.**

So Negatives just go the other way.

### Example: What is 7.1 × 10^{-3} ?

Well, it is really 7.1 x (^{1}/_{10} × ^{1}/_{10} × ^{1}/_{10}) = 7.1 × 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 |

## Try It Yourself

Enter a number and see it in Scientific Notation:

Now try to use Scientific Notation yourself:

## Summary

The index of 10 says how many places to move the decimal point. Positive means move it to the right, negative means to the left. Example:

Number | In ScientificNotation | In Words | |

Positive Powers | 5,000 | 5 × 10^{3} | 5 Thousand |

Negative Powers | 0.005 | 5 × 10^{-3} | 5 Thousandths |