Exponents

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

8 to the Power 2

In 82 the "2" says to use 8 twice in a multiplication,
and then 8ii = 8 × eight = 64

In words: 8two could be called "8 to the power 2" or "8 to the second power", or simply "8 squared"

Exponents are also chosen Powers or Indices.

Some more examples:

Instance: five3 = v × five × five = 125

  • In words: 53 could be chosen "5 to the third power", "5 to the ability 3" or simply "5 cubed"

Instance: 24 = 2 × 2 × 2 × 2 = 16

  • In words: two4 could be called "ii to the fourth power" or "2 to the ability 4" or simply "two to the 4th"

Exponents make it easier to write and utilise many multiplications

Case: 96 is easier to write and read than 9 × ix × nine × 9 × 9 × 9

You lot can multiply any number by itself every bit many times as y'all desire using exponents.

Try here:

algebra/images/exponent-calc.js

So in general:

anorthward tells you to multiply a by itself,
then in that location are due north of those a's:		 exponent definition

Another Mode of Writing It

Sometimes people apply the ^ symbol (to a higher place the 6 on your keyboard), as it is easy to type.

Example: 2^iv is the same every bit 2iv

  • 2^4 = 2 × 2 × 2 × 2 = sixteen

Negative Exponents

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

And so nosotros divide past the number each time, which is the same every bit multiplying by 1 number

Instance: 8-1 = 1 8 = 0.125

We tin go along on similar this:

Example: 5-three = 1 5 × 1 5 × one five = 0.008

Only it is often easier to practise it this way:

5-3 could also be calculated like:

1 5 × five × 5 = 1 v3 = 1 125 = 0.008

Negative? Flip the Positive!

negative-exponent

That last case showed an easier way to handle negative exponents:

  • Calculate the positive exponent (an )
  • So take the Reciprocal (i.e. 1/an )

More than Examples:

Negative Exponent Reciprocal of
Positive Exponent		 Answer
4-2 = 1 / 42 = one/16 = 0.0625
ten-3 = ane / tenthree = one/one,000 = 0.001
(-2)-three = 1 / (-2)iii = 1/(-eight) = -0.125

What if the Exponent is 1, or 0?

one If the exponent is i, so you lot just have the number itself (instance 91 = ix)
0 If the exponent is 0, and then y'all go one (example 90 = 1)
But what about 00 ? Information technology could be either one or 0, then people say information technology is "indeterminate".

Information technology All Makes Sense

If you lot wait at that table, you will see that positive, zero or negative exponents are really role of the same (adequately simple) pattern:

Case: Powers of 5
.. etc.. exponent 5 times larger or smaller
52 5 × five 25
vi 5 v
50 1 1
v-1 1 5 0.2
v-2 1 five × 1 5 0.04
.. etc..

Be Careful About Grouping

To avert confusion, use parentheses () in cases like this:

With () : (−2)2 = (−ii) × (−two) = 4
Without () : −2two = −(two2) = −(2 × 2) = −four

With () : (ab)2 = ab × ab
Without () : ab2 = a × (b)ii = a × b × b

305, 1679, 306, 1680, 1077, 1681, 1078, 1079, 3863, 3864