In this chapter, you will learn about the short cuts to writing
$2\times 2\times 2\times 2$ . This is known as writing a number in
exponential notation .
Definition
Exponential notation is a short way of writing the same number multiplied by
itself many times. For example, instead of
$5\times 5\times 5$ , we write
${5}^{3}$ to show that the number 5 is multiplied by itself 3 times and we say “5 to the power of 3”. Likewise
${5}^{2}$ is
$5\times 5$ and
${3}^{5}$ is
$3\times 3\times 3\times 3\times 3$ . We will now have a closer look at writing numbers using exponential notation.
Exponential Notation
Exponential notation means a number written like
$${a}^{n}$$
where
$n$ is an integer and
$a$ can be any real number.
$a$ is called the
base and
$n$ is called the
exponent or
index .
If
$n$ is an even integer, then
${a}^{n}$ will always be positive for any non-zero real number
$a$ . For example, although
$-2$ is negative,
${(-2)}^{2}=-2\times -2=4$ is positive and so is
${(-2)}^{-2}=\frac{1}{-2\times -2}=\frac{1}{4}$ .
Laws of exponents
There are several laws we can use to make working with exponential numbers easier. Some of these laws might have been seen in earlier grades, but we will list all the laws here for easy reference and explain each law in detail, so that you can understand them and not only remember them.
This simple law is the reason why exponentials were originally invented. In the days before calculators, all multiplication had to be done by hand with a pencil and a pad of paper. Multiplication takes a very long time to do and is very tedious. Adding numbers however, is very easy and quick to do. If you look at what this law is saying you will realise that it means that adding the exponents of two exponential numbers (of the same base) is the same as multiplying the two numbers together. This meant that for certain numbers, there was no need to actually multiply the numbers together in order to find out what their multiple was. This saved mathematicians a lot of time, which they could use to do something more productive.
Application using exponential law 2:
${a}^{m}\times {a}^{n}={a}^{m+n}$
${x}^{2}\xb7{x}^{5}$
${2}^{3}.{2}^{4}$ [Take note that the base (2) stays the same.]
$3\times {3}^{2a}\times {3}^{2}$
Exponential law 3:
${a}^{-n}=\frac{1}{{a}^{n}},\phantom{\rule{1.em}{0ex}}a\ne 0$
Our definition of exponential notation for a negative exponent shows that
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Adu
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=