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LSC2: INDICES

This unit is about indices
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Index (Power)

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Index (Power)

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

It is written as a small number to the right and above the base number.

In this example: 82 = 8 × 8 = 64

The plural of index is indices.

(Other names for index are exponent or power.)

Index Notation and Powers of 10

10 to the Power 2

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

102 means 10 × 10 = 100

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

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

  • In words: 104 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 × 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) like to write numbers this way.

Example: The Mass of the Sun

The Sun has a Mass of 1.988 × 1030 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 × 1015 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 × 104

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

calculator e notation

Calculators often use “E” or “e” like this:

Example: 6E+5 is the same as 6 × 105

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

Example: 3.12E4 is the same as 3.12 × 104

  • 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 × 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 arrow 13.5 arrow 135. arrow 1350. arrow 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 arrow 0.71 arrow 0.071 arrow 0.0071

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 Scientific
Notation
 In Words
Positive Powers 5,000 5 × 103 5 Thousand
Negative Powers 0.005 5 × 10-3 5 Thousandths

Laws of Exponents

Exponents are also called Powers or Indices

8 to the Power 2

The exponent of a number says how many times to use the number in a multiplication.

In this example: 82 = 8 × 8 = 64

In words: 82 could be called “8 to the second power”, “8 to the power 2” or simply “8 squared”

Try it yourself34 = 3 × 3 × 3 × 3 = 81

So an Exponent saves us writing out lots of multiplies!

Example: a7

a7 = a × a × a × a × a × a × a = aaaaaaa

Notice how we wrote the letters together to mean multiply? We will do that a lot here.

Example: x6 = xxxxxx

 

The Key to the Laws

Writing all the letters down is the key to understanding the Laws

Example: x2x3 = (xx)(xxx) = xxxxx = x5

Which shows that x2x3 = x5, but more on that later!

So, when in doubt, just remember to write down all the letters (as many as the exponent tells you to) and see if you can make sense of it.

All you need to know …

The “Laws of Exponents” (also called “Rules of Exponents”) come from three ideas:

pencil paper The exponent says how many times to use the number in a multiplication.
turn over A negative exponent means divide, because the opposite of multiplying is dividing
pie slice
A fractional exponent like 1/n means to take the nth root: x^(1/n) = n-th root of x

 

If you understand those, then you understand exponents!

And all the laws below are based on those ideas.

Laws of Exponents

Here are the Laws (explanations follow):

Law Example
x1 = x 61 = 6
x0 = 1 70 = 1
x-1 = 1/x 4-1 = 1/4
xmxn = xm+n x2x3 = x2+3 = x5
xm/xn = xm-n x6/x2 = x6-2 = x4
(xm)n = xmn (x2)3 = x2×3 = x6
(xy)n = xnyn (xy)3 = x3y3
(x/y)n = xn/yn (x/y)2 = x2 / y2
x-n = 1/xn x-3 = 1/x3
And the law about Fractional Exponents:
x^(m/n) = n-th root of (x^m) = (n-th root of x)^m x^(2/3) = 3rd root of (x^2) = (3rd root of x)^2

Laws Explained

The first three laws above (x1 = x, x0 = 1 and x-1 = 1/x) are just part of the natural sequence of exponents. Have a look at this:

Example: Powers of 5
.. etc.. exponent 5x larger smaller
52 1 × 5 × 5 25
51 1 × 5 5
50 1 1
5-1 1 ÷ 5 0.2
5-2 1 ÷ 5 ÷ 5 0.04
.. etc..

Look at that table for a while … notice that positive, zero or negative exponents are really part of the same pattern, i.e. 5 times larger (or 5 times smaller) depending on whether the exponent gets larger (or smaller).

The law that xmxn = xm+n

With xmxn, how many times do we end up multiplying “x”? Answer: first “m” times, then by another “n” times, for a total of “m+n” times.

Example: x2x3 = (xx)(xxx) = xxxxx = x5

So, x2x3 = x(2+3) = x5

The law that xm/xn = xm-n

Like the previous example, how many times do we end up multiplying “x”? Answer: “m” times, then reduce that by “n” times (because we are dividing), for a total of “m-n” times.

 

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