Math Operations on Integers, Decimals, and Percentages

Study Notes

Integers:
- Addition: commutative and associative
- Subtraction: not commutative, but associative
- Multiplication: commutative and associative
- Division: not commutative, not associative
Decimals:
- Operations same as integers, but with decimal points
- Convert fractions to decimals by dividing numerator by denominator
Percentages:
- Increase: multiply by (1 + percentage/100)
- Decrease: multiply by (1 - percentage/100)
- Change from one quantity to another: ((new - old)/old) * 100
Ratios:
- Compare two quantities: a:b or a/b
- Equivalent ratios: multiplying or dividing both terms by the same number
Real Numbers:
- Include all integers, decimals, and fractions
- Can be represented on the number line

Polynomials:
- Addition: combine like terms
- Subtraction: combine like terms with opposite signs
- Multiplication: distributive property
- Division: long division or synthetic division
Algebraic Expressions:
- Simplify: combine like terms and cancel out any common factors
- Evaluate: substitute values for variables

Factoring out the greatest common factor (GCF):
- Find the GCF of all terms
- Divide each term by the GCF
Factoring by grouping:
- Group terms with common factors
- Factor out the common factor
Factoring quadratic expressions:
- x^2 + bx + c = (x + d)(x + e)
- Find d and e by factoring c and b

Simplifying rational expressions:
- Factor numerator and denominator
- Cancel out common factors
Adding and subtracting rational expressions:
- Find the least common multiple (LCM) of the denominators
- Convert each expression to have the LCM as the denominator
Multiplying and dividing rational expressions:
- Multiply or divide the numerators and denominators separately

Exponents:
- Rules: product of powers, power of a product, and power of a power
- Negative exponents: 1/x^n
Rational Exponents:
- a^(m/n) = nth root of a^m
- Rules: product of powers, power of a product, and power of a power
Radicals:
- Square root: √a
- Cube root: ³√a
- Simplify: simplify the expression inside the radical

Rearranging formulas:
- Isolate the desired variable by performing inverse operations
- Use algebraic properties to simplify the formula

Integers:
- Have commutative and associative properties for addition and multiplication
- Subtraction is not commutative, but is associative
- Division is not commutative and not associative
Decimals:
- Follow the same operations as integers, with consideration of decimal points
- Can be converted from fractions by dividing the numerator by the denominator
Percentages:
- Calculate increases by multiplying by (1 + percentage/100)
- Calculate decreases by multiplying by (1 - percentage/100)
- Find percentage changes by ((new - old)/old) * 100
Ratios:
- Compare two quantities using a:b or a/b notation
- Equivalent ratios can be found by multiplying or dividing both terms by the same number
Real Numbers:
- Include all integers, decimals, and fractions
- Can be represented on a number line

Polynomials:
- Add by combining like terms
- Subtract by combining like terms with opposite signs
- Multiply using the distributive property
- Divide using long division or synthetic division
Algebraic Expressions:
- Simplify by combining like terms and canceling out common factors
- Evaluate by substituting values for variables

Factoring out the GCF:
- Find the greatest common factor of all terms
- Divide each term by the GCF
Factoring by grouping:
- Group terms with common factors
- Factor out the common factor
Factoring quadratic expressions:
- Express as x^2 + bx + c = (x + d)(x + e)
- Find d and e by factoring c and b

Simplifying rational expressions:
- Factor the numerator and denominator
- Cancel out common factors
Adding and subtracting rational expressions:
- Find the least common multiple of the denominators
- Convert each expression to have the LCM as the denominator
Multiplying and dividing rational expressions:
- Multiply or divide the numerators and denominators separately

Exponents:
- Follow the rules of product of powers, power of a product, and power of a power
- Negative exponents represent reciprocals: 1/x^n
Rational Exponents:
- Represented as a^(m/n) = nth root of a^m
- Follow the rules of product of powers, power of a product, and power of a power
Radicals:
- Represented as square root (√a) or cube root (³√a)
- Simplify by simplifying the expression inside the radical