Simplification Formulas
Basic Arithmetic Operations
The basic arithmetic operations include addition, subtraction, multiplication, and division.
Addition
\[
a + b = b + a
\]
Subtraction
\[
a - b \neq b - a
\]
Multiplication
\[
a \times b = b \times a
\]
Division
\[
\frac{a}{b} \neq \frac{b}{a}
\]
BODMAS/BIDMAS Rule
The BODMAS/BIDMAS rule is used to determine the order of operations in a mathematical expression.
- Brackets
- Orders (i.e., powers and roots, etc.)
- Division and Multiplication (left to right)
- Addition and Subtraction (left to right)
Fraction Operations
Adding Fractions
\[
\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}
\]
Subtracting Fractions
\[
\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}
\]
Multiplying Fractions
\[
\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}
\]
Dividing Fractions
\[
\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}
\]
Square and Cube Roots
Square Root
\[
\sqrt{a}
\]
Cube Root
\[
\sqrt[3]{a}
\]
Exponents and Powers
Exponentiation
\[
a^b = a \times a \times \cdots \times a \quad (\text{b times})
\]
Power of a Power
\[
(a^m)^n = a^{mn}
\]
Multiplying Powers with Same Base
\[
a^m \times a^n = a^{m+n}
\]
Dividing Powers with Same Base
\[
\frac{a^m}{a^n} = a^{m-n}
\]
Logarithms
Logarithm Definition
\[
\log_b(a) = c \iff b^c = a
\]
Product Rule
\[
\log_b(xy) = \log_b(x) + \log_b(y)
\]
Quotient Rule
\[
\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)
\]
Power Rule
\[
\log_b(x^y) = y \log_b(x)
\]