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Monday, November 8, 2010

The Scourge and Bane - Exponentials

I am sure many of you hate the sordid looking Indices and Roots. But of course, I don't blame you. Really, who could stand





much less be comfortable solving it. But the sad truth is that Indices and Roots are basics that we should be experts in - whether we are working on Number Systems, Algebra or even Geometry! So, let's get crackin'

Warning: I will just discuss the basic rules in this post for people who dislike exponents.

Remember 'The Four Prongs' post? There we discussed how 34 is just 3 x 3 x 3 x 3. - The '3' is called the base and the '4' is called its index (plural - indices). It follows then that 32 would be just 3 x 3. So now, if I multiply 34 by 32, what do I get? 3 x 3 x 3 x 3 x 3 x 3 = 36 . The indices just got added!

Rule 1: am × an = am + n

Now tell me what happens when I divide
34 by 32 ?

It looks something like the following:




But we know that 3 x 3 =
32 . So essentially, 34 /32 = 34 – 2 = 32

Rule 2:
am / an = am - n
When you divide two numbers with the same base, the index of the divisor gets subtracted from the index of the dividend.

Now that we have discussed Multiplication and Division, we need to think about Addition and Subtraction!
What happens when I add
34 to 32 ? What is 34 + 32 ? Can I still add indices? Think. 3 x 3 x 3 x 3 + 3 x 3 = 81 + 9 = 90. I cannot play with the indices when dealing with Addition and Subtraction. The best I can do is take out a common factor. e.g. 34 + 32 = 32 (32 + 1) = 9.10 = 90
I have taken two 3s out from both the numbers and added the rest. Saves me time I need to calculate 3 x 3 x 3 x 3. Exact same thing can be done for Subtraction too.

Now, what is
3-4 ? It is extremely easy to handle negative indices. Just flip the base and the index becomes positive. So 3-4 is 1/34 .
This implies the following:
  • 1/3-4 = 34
  • 34 = 1/3-4
  • 1/34 = 3-4

When you want to change the sign of the index, flip the base. When you want to flip the base, change the sign of the index!

Rule 3: For any number a, a0 = 1
e.g. 30 = 1

How about a quick question now?
Given:



What should replace the ?? to make the equation valid?
I know 4 can be written as 2^2 and I can flip the base in the denominator to the numerator, thereby changing the sign of the index.
We get:

2?? . 22.2-4.2-2 = 1

Here, all the terms are multiplied and there bases are same i.e. 2, so

the indices should be added.

2??+2-4-2 = 1 = 20

Question: From where did we get 20 ? We know, 20 = 1 so if we have 1, we can write it as 20.

That is the only way in which a term with base 2 could have become 1.

Now, since the bases on both sides of the equation are same, the

indices should also be the same.

?? + 2 - 4 - 2 = 0

?? = 4

Note: Of course, you could have cancelled off 4 in the numerator with 22 in the denominator and got your answer directly, but here we are learning to use the various rules of exponents.




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