Math Operations

Math Operations

Operations with Parentheses

Perform multiplications and divisions first, then additions and subtractions. When simplifying fractions, the greatest common divisor should be used to reduce both the numerator and the denominator.

Expressing Fractions as Decimals

When the decimal is purely periodic (e.g., n = 235.2): The numerator is the number without the decimal point minus the non-periodic part, and the denominator is as many 9s as there are figures in the period.

When the decimal is mixed periodic (e.g., 1.372): The numerator is the number without the decimal point minus the non-periodic part. The denominator consists of as many 9s as there are periodic figures followed by as many 0s as there are non-periodic figures after the decimal point.

Powers and Roots

Properties of Powers:

1. Anything raised to the power of 0 is equal to 1.
2. a^b x a^c = a^b+c
3. a^b x b^c = a^b x b^c
4. (a^b)^c = a^bxc
5. a^-b = 1/a^b
6. 1/a^-b = a^b
7. (a x b)^c = a^c x b^c
8. (a/b)^c = a^c/b^c
9. (a/b)^-c = b^c/a^c

Problem-Solving Example

We have 90 meters of wire. We sell 2/3 at €3 per meter, 1/6 at €4 per meter, and the rest at €2 per meter. How much money did we earn if we bought the wire at €2 per meter?

Solution:

1. 2/3 of 90 = 60 meters; 30 meters remaining
2. 1/6 of 30 = 5 meters; 25 meters remaining
3. First stage: 60 meters x €3/meter = €180
4. Second stage: 5 meters x €4/meter = €20
5. Remaining: 25 meters x €2/meter = €50
6. Total earnings: €180 + €20 + €50 = €250
7. Cost: 90 meters x €2/meter = €180
8. Profit: €250 - €180 = €70

Polynomial Division

First, divide the term of the highest degree in the dividend by the term of the highest degree in the divisor. Then, multiply the result by the divisor and subtract it from the dividend. Repeat this process until the remainder has a degree less than the divisor.

Scientific Notation

Express a number with one significant figure (other than 0) before the decimal point.

Intervals

A closed interval [a, b] includes the points a and b. An open interval (a, b) does not include the points a and b.

Notable Identities

(a + b)² = a² + 2ab + b²
(-a + b)² = a² - 2ab + b²
(a - b)² = a² - 2ab + b²
(-a - b)² = a² + 2ab + b²
(a + b)(a - b) = a² - b²

Quadratic Formula

x = [-b ± √(b² - 4ac)] / 2a