10 Mar 2013

What are Indices or 'Powers'? Numbers Squared, Cubed, and more.

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What are these? [and how did you teach them?]

1²  2²  3²  4²   5²    6²    7²    8²   9²    10² 

They're the numbers up to ten 'to the power of 2' or 'squared'. 
1² = 1x1  answer: 1² = 1

2² = 2x2  answer: 2² = 4     Beware:
3² = 3x3  answer: 3² = 9     NOT 3x2=6
4² = 4x4  answer: 4² = 16   NOT 4x2=8
5² = 5x5  answer: 5² = 25   NOT 5x2=10
so

1²  2²  3²  4²   5²    6²    7²    8²   9²    10²    11²    12²
2   4   9  16  25    36   49   64   81   100   121   144

Number (n), is multiplied by itself: n x n = n² 

The funny little ² jumping up above the line is called an index. The numbers on the line are called 'the base'.
so in 3², 3 is the base, 2 is the index.

So far so good. We've done 'to the power of 2'. So how many other 'powers', or indices are there? Is there a power of 3?
Yes.

Any number can be an 'index'.
  n⁴  n⁵  n⁶  n⁷ and so on.
The number on the line (n in the example above) is called the base. e.g.
in 4², 4 is the base, 2 is the index (or power)
in 6³, 6 is the base, 3 is the index (or power)

Any number can be a base, so we can have: 2³, 54³, 100³, 1034³ or
4⁴ 23⁴ 191⁴ 223⁴ 9872⁴ and so on.

If a number is raised to the power of 3 (index 3), what do we do?
 = n x n x n (n times by itself three times: NOT the same as n x 3)
This is also known as 'cubed'. So 4x4x4 is 4 cubed, or 4³ = 64
Try it on a calculator, or, if you want to do it in your head, find 4 x 4 first (=16) then multiply that, again, by 4. So 16 x 4 = 64.

If a number is raised to the power of 4 (index 4), what do we do?
n⁴= n x n x n x n (n multiplied by itself four times: NOT the same as n x 4)
So 4⁴ = 4 x 4 x 4 x 4 = 256
Try it on a calculator, or, if you want to do it in your head, find 4 x 4 first (=16) then multiply that, again, by 4. So 16 x 4 = 64. Then multiply it again by 4. So 64 x 4 = 256

If a number is raised to the power of 5 (index 5), what do we do?
n⁵= n x n x n x n x n (n multiplied by itself five times: NOT the same as n x 5)
So 4 = 4 x 4 x 4 x 4 x 4 = 1024
Try it on a calculator, or, if you want to do it in your head, find 4 x 4 first (=16) then multiply that, again, by 4. So 16 x 4 = 64. Then multiply it again by 4. So 64 x 4 = 256. Then multiply again, by 4. So 256 x 4 = 1024.

If a number is raised to the power of 5 (index 5), what do we do?
n⁶n x n x n x n x n x n (n multiplied by itself six times: NOT the same as n x 6)
Try it on a calculator, or, if you want to do it in your head, find 4 x 4 first (=16) then multiply that, again, by 4. So 16 x 4 = 64. Then multiply it again by 4. So 64 x 4 = 256. Then multiply again, by 4. So 256 x 4 = 1024. Multiply once more, by 4. So 1024 x 4 = 4196.