Exponentials

We often know with number that are repeatedly multiplied together. In Mathematic, that be said indices or exponents to represent such expressions. For example, 5 x 5 x 5 = 5³.

Indices have many applications in areas such ac finance, engineering, physics, electronics, biology, and computer science. Problem in exponents may involve situations where quantities increase or decrease over time. Such problems are often examples of exponential growth or decay.

• Index Notation

Rather than writing 3 x 3 x 3 x 3 x 3, we can write such a product as 3­_­­⁵. 3⁵ reads “three to the power of five” or “ three with index five”.

thus 4³ = 4 x 4 x 4 and 5⁶ = 5 x 5 x 5 x 5 x 5 x 5

if n is positive integer,then an is the product from a with n factors, so: an = a x a x a x a x . . . . x a with a as many as n factors

• Negative Base

In discussion before, the examples only index from positive base. We will now briefly look at negative bases. Consider the statements below:

(-1)1 = -1	(-2)1 = -2
(-1)2 = -1 x -1 = 1	(-2)2 = -2 x -2 = 4
(-1)3 = -1 x -1 x -1 = -1	(-2)3 = -2 x -2 x -2 = -8
(-1)4 = -1 x -1 x -1 x -1 = 1	(-2)4 = -2 x -2 x -2 x -2 = 16

From the patterns above we can see that

a.       A negative base raised to an odd power is negatif

b.      A negative base raised to an even power is positive

• Index Law

The following are laws of indices for  m, n Î Z :

index Law	Information
am x an = am+n	To multiply numbers with the same base, keep the base and add the indices.
(am)/(an) = am-n , a ≠ 0	To devide numbers with the same base, keep the base and subtract the indices.
(am)n = am x n	When raising a power to a power, keep the base and multiply the indices.
(ab)n = anbn	The power of a product is the product of the powers.
(a/b)n = an/bn, b ≠ 0	The power of a quotient is the quotient of the powers.
a0 = 1, a ≠ 0	Any non-zero number raised to the power of zero is 1.
a-n = 1/an dan 1/a-n = an