Tetration

This can be calculated in several ways.

1. By multiplication of fractions

Remember that all whole numbers k can be written in the form k/1. We multiply the numerator (top) by numerator and denominator by denominator, and get:

$$\frac{1}{6}*\frac{18}{1}= \frac{18}{6} = 3$$

2. By division of the whole number

To multiply a number with 1/x is the same as dividing it with x. We get:

$$\frac{1}{6}*18 = \frac{18}{6}= 3$$

For completeness I might add this: If you wish to multiply a whole number with a fraction where the numerator (top) is unlike 1, you simply multiply the whole number with the numerator, and divide the product with the denominator:

$$\frac{a}{b}* k = \frac{a*k}{b}$$

22/7 = 3,142857(142857)...

As you see this fraction can be represented as a repeating decimal with reptend (repeting sequence) 142857. 

This fraction is a good approximation for the number pi. 

Percent means "per hundred"

5% = 5/100 

5% of something means 5% * something.

So, we get:

5% * 125 000 = 5/100 * 125 000 =  6250 

How many 75 ml bottles can be filled by 7 1/2 litres

First we transform all the units into litres: 

We know that milli means "one thousandth", i.e. 1/1000. Therefore 75 mL = 75L * 1/1000 = 75/1000 L = 0,075L

Also:

$$7 \frac{1}{2}L= 7L + \frac{1}{2}L =\frac{14}{2}L + \frac{1}{2}L = \frac{15}{2}L$$

So, how many times can you pour 75 mL in order to fill 7 1/2 L ?

Remember that when we divide one fraction by another, we multiply the first one by the multiplicative inverse of the last one, we get:

$$\frac{15}{2}L/\frac{75}{1000}L=\frac{15}{2}/\frac{75}{1000}= \frac{15}{2}*\frac{1000}{75}= 100$$

We see that the units cancels out and we get a dimentionless quantity, which is what is expected in this case.

Order of Operations:

1. Solve the paranthesis

2. Multiply (*)   /    Divide (:)

3. Add (+)    /   Subtract    (-)

(2⋅52 - 72)1714 = (104 - 72)1714 =  32*1714 = <Your answer>

If you have a usual French deck consisting of 52 different card, and want to know how many different ways to list all the cards, you can use factorial.

The total # of different ways to list all the 52 cards, such that you have a 1st card, 2nd card and so forth, is 52!

52! = 52*51*50*49* ... *4*3*2*1 ≈ 8,1 *10⁶⁷.

Why is this?

Your 1st card have to be one of the 52 cards: 52 possibilites.

Your 2nd card have to be one of the remaining 51 cards: a total of 52*51 possibilities.

Your 3rd card have to be one of the remaining 50 cards: a total of 52*51*50 possibilities.

Your 4th card have to be one of the remaining 49 cards: a total of 52*51*50*49 possibilities.

And so on until there is a single card left.

Scientific notation is in the form: a * 10^b.

We count the number of digits: 10. We know that the exponent is one less than the # of digits. Therefore it's going to be on the form: a * 10⁹. 

We get rid of all the trailing zeros, and put a period after the first digit: 1.205

So the # in scientific form is: 1.205 * 10⁹.

When you divide one fraction by another, you multiply the former with the inverse of the latter. 

$$\frac{a}{b} : \frac{c}{d} = \frac{a}{b} * \frac{d}{c}$$

First we convert grams into kilos. 

125g = 0,125 Kg

Remember that the gravity of Earth is ≈ 9,81 m/s².

We use this formula:

E_p = mgh = 0,125 Kg * 9,81 m/s² * 3m ≈ 3,68 Kg m²/s² 

Check: J =  Kg m²/s²    

f[g(x)] = f(-2x + 2) = 4(-2x + 2) - 3  = -8x + 8 - 3 = -8x + 5

g[f(x)] = g(4x - 3) = -2(4x - 3) + 2 = 2(3 - 4x) + 2          [ since -(a - b) = b - a ]

g[f(x)] = -8x + 6 + 2 = -8x + 8