Tetration

P: Pepperoni

M: Mushroom

U: one or the other or both

∩: Both at the same time

P U M = 12       P = 5      M = 8

A formula (that I got from probability theory) says that:

A ∩ B = A + B - A U B 

In our case:

P ∩ M = P + M - P U M = 5 + 8 - 12 = 1 

So the pizza consists of 4 slices w/ only pepperoni, 7 slices w/ only mushroom, and 1 slice w/ both. 

298 K is just 25 ^oC, or "room temperature".

[H⁺] = 5.35 * 10^-3 M 

We know that:

[H⁺][OH^-] = 10^-14 M²

[OH^-] = 10^-14 M²/[H⁺] ≈ 1.87 * 10^-12 M

To divide a fraction by a number is the same as to multiply the fraction by the multiplicative inverse of that number, i.e:

$$\frac{a}{b}/c = \frac{a}{b}*\frac{1}{c}$$

We get:

$$\frac{5}{3}/9 = \frac{5}{3}*\frac{1}{9}$$

We multiply (*) numerator (top) by numerator and denominator (bottom) by denominator:

$$\frac{5}{3}*\frac{1}{9} = \frac{5 * 1}{3*9}= \frac{5}{27}$$

42.7²

This formula might be helpful:

(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca = a² + b² + c² + 2(ab + bc + ca)

We get:

42.7² = (40 + 2 + .7)² =

40² + 2² + .7² + 2(40 * 2 + 2 * .7 + 40 * .7) = 

1600 + 4 + .49 + 2(80 + 1.4 + 28) = 

1604.49 + 2 * 109.4 = 1823.29 

Pi is approximately 3.14. 

Pi is a mathematical constant. It's definition is the ration between a circles circumference and it's diameter:

Pi = C/d

Pi is an irrational number, i.e. it cannot be written as a fraction of two integer (An integer is a positive or negative whole number or zero). Since it is an irrational number, it follows that pi has infinitely many decimal digits, and that pi cannot be represented as terminating or repeating decimals. 

... 100 times as many.

100 * 6.022 * 10²³ = 6.022 * 10² * 10²³ = 6.022 * 10²⁵

Percent means "per hundred". 5% = 5/100 = 1/20.

"7% of something" means "7% * something"

260% of 7 is :

260% * 7 = 7 * 260/100 = 18.2

20j² - 87 = -82      Add 87 to both sides:

20j² = 5      Divide both sides by 20:

j² = 1/4     Take the square root of both sides:

j = 1/2

3(3a - 1) + 2     Solve the paranthesis:

9a - 3 + 2     Do the subtraction:

9a - 1 

[18 – (10 – 6)] + 1  

First get rid of the unneccesary paranthesis:

18 – (10 – 6) + 1      

EITHER: Remember that -(a - b) = b - a

18 + 6 - 10 + 1 = 15

OR: Solve the usual way, be VERY carful so you avoid carless mistakes based on multiple minus signs:

18 – (10 – 6) + 1  = 18 - 4 + 1 = 15