Math Operations on Integers, Decimals, and Percentages

Operations on Integers, Decimals, Percentages, Ratios, and Real Numbers

• 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

Operations on Polynomials and Algebraic Expressions

• 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

• 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

Rational Expressions

• 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, Rational Exponents, and Radicals

• 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

Transposition of Formulae

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

Operations on Numbers

• 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

Algebraic Operations

• 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 Techniques

• 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

Rational Expressions

• 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 and Radicals

• 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

Transposition of Formulae

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