Math Classroom 20170115

The post summarized the topics:

  1. Laws of Exponents
  2. Exponents of 10
  3. Exponents of 2
  4. Exponents as fractions

Laws of Exponents

Law Example
x1 = x 61 = 6
x0 = 1 70 = 1
x-1 = 1/x 4-1 = 1/4
xmxn = xm+n x2x3 = x2+3 = x5
xm/xn = xm-n x6/x2 = x6-2 = x4
(xm)n = xmn (x2)3 = x2×3 = x6
(xy)n = xnyn (xy)3 = x3y3
(x/y)n = xn/yn (x/y)2 = x2 / y2
x-n = 1/xn x-3 = 1/x3

Laws Explained

The first three laws above (x1 = x, x0 = 1 and x-1 = 1/x) are just part of the natural sequence of exponents. Have a look at this:

Example: Powers of 5
  .. etc..  
52 1 × 5 × 5 25
51 1 × 5 5
50 1 1
5-1 1 ÷ 5 0.2
5-2 1 ÷ 5 ÷ 5 0.04
  .. etc..  

Look at that table for a while … notice that positive, zero or negative exponents are really part of the same pattern, i.e. 5 times larger (or 5 times smaller) depending on whether the exponent gets larger (or smaller).

The law that xmxn = xm+n

With xmxn, how many times do we end up multiplying “x”? Answer: first “m” times, then by another “n” times, for a total of “m+n” times.

Example: x2x3 = (xx)(xxx) = xxxxx = x5

So, x2x3 = x(2+3) = x5

The law that xm/xn = xm-n

Like the previous example, how many times do we end up multiplying “x”? Answer: “m” times, then reduce that by “n” times (because we are dividing), for a total of “m-n” times.

Example: x4/x2 = (xxxx) / (xx) = xx = x2

So, x4/x2 = x(4-2) = x2

(Remember that x/x = 1, so every time you see an x “above the line” and one “below the line” you can cancel them out.)

This law can also show you why x0=1 :

Example: x2/x2 = x2-2 = x0 =1

The law that (xm)n = xmn

First you multiply “m” times. Then you have to do that “n” times, for a total of m×n times.

Example: (x3)4 = (xxx)4 = (xxx)(xxx)(xxx)(xxx) = xxxxxxxxxxxx = x12

So (x3)4 = x3×4 = x12

The law that (xy)n = xnyn

To show how this one works, just think of re-arranging all the “x”s and “y”s as in this example:

Example: (xy)3 = (xy)(xy)(xy) = xyxyxy = xxxyyy = (xxx)(yyy) = x3y3

The law that (x/y)n = xn/yn

Similar to the previous example, just re-arrange the “x”s and “y”s

Example: (x/y)3 = (x/y)(x/y)(x/y) = (xxx)/(yyy) = x3/y3

Exponents of 10

“Exponents of 10” is a very useful way of writing down large or small numbers.

Instead of having lots of zeros, you show how many powers of 10 will make that many zeros
Example: 5,000 = 5 × 1,000 = 5 × 103

5 thousand is 5 times a thousand. And a thousand is 103. So 5 times 103 = 5,000

Can you see that 103 is a handy way of making 3 zeros?
Scientists and Engineers (who often use very big or very small numbers) like to write numbers this way.
Example: The Mass of the Sun

The Sun has a Mass of 1.988 × 1030 kg.

It is too hard to write 1,988,000,000,000,000,000,000,000,000,000 kg

(And very easy to make a mistake counting the zeros!)
Example: A Light Year (the distance light travels in one year)

It is easier to use 9.461 × 1015 metres, rather than 9,461,000,000,000,000 metres

Sometimes people use the ^ symbol (above the 6 on your keyboard), as it is easy to type.
Example: 3 × 10^4 is the same as 3 × 104
•3 × 10^4 = 3 × 10 × 10 × 10 × 10 = 30,000

calculator e notation

Calculators often use “E” or “e” like this:

Example: 6E+5 is the same as 6 × 105
•6E+5 = 6 × 10 × 10 × 10 × 10 × 10 = 600,000
Example: 3.12E4 is the same as 3.12 × 104
•3.12E4 = 3.12 × 10 × 10 × 10 × 10 = 31,200

The index of 10 says …

… how many places to move the decimal point to the right.

Negative Exponents of 10

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

A negative power means how many times to divide by the number.
Example: 5 × 10-3 = 5 ÷ 10 ÷ 10 ÷ 10 = 0.005

Just remember for negative powers of 10:

For negative powers of 10, move the decimal point to the left.

Exponents of 2

Here is a list of the number 2 raised to the power of every number from 0 to 100. Why did I make this? Because I have WAY too much time on my hands! You’re probably wondering what good this is. Well aside from the obvious (It allows quick look-up to 2’s powers) It can also be used to find out how many colors will be displayed on a screen. Example: If you’re computer monitor is set to display 16-bit color, look at 2 raised to the 16th power, and you’ll see that the value is 65,536. That means that at your current display, you can have 65,536 colors on the screen at once. Now doesn’t that make you feel all warm and mushy inside?

Exponents of 2  Value
0 1
1 2
2 4
3 8
4 16
5 32
6 64
7 128
8 256
9 512
10 1,024
11 2,048
12 4,096
13 8,192
14 16,384
15 32,768
16 65,536

 

Exponents as fractions

what if the exponent is a fraction?

An exponent of 1/2 is actually square root

And an exponent of 1/3 is cube root

An exponent of 1/4 is 4th root

And so on!

exponent fractional 1

General Rule

It worked for ½, it worked with ¼, in fact it works generally:

x1/n = The n-th Root of x

What about a fractional exponent like 43/2 ?

That is really saying to do a cube (3) and a square root (1/2), in any order.

Let me explain.

A fraction (like m/n) can be broken into two parts:

  • a whole number part (m) , and
  • a fraction (1/n) part

So, because m/n = m × (1/n) we can do this:

x^(m/n) = x^(1/n by m) = (x^(1/n))^m = (nth root of x)^m

The order does not matter, so it also works for m/n = (1/n) × m:

x^(m/n) = x^(1/n by m) = (x^(1/n))^m = (nth root of x)^m

And we get this:

pie slice
A fractional exponent like m/n means:

  Do the m-th power, then take the n-th root

OR Take the n-th root and then do the m-th power

x^(m/n) = n-th root of (x^m) = (n-th root of x)^m

 

Some examples:

Example: What is 43/2 ?

43/2 = 43×(1/2) = √(43) = √(4×4×4) = √(64) = 8

or

43/2 = 4(1/2)×3 = (√4)3 = (2)3 = 8

Either way gets the same result.

Example: What is 274/3 ?

274/3 = 274×(1/3) = cube root(274) = cube root(531441) = 81

or

274/3 = 27(1/3)×4 = (cube root27)4 = (3)4 = 81

 

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