In Depth Notes on Quantitative Methods (QMS260s) - Unit 2: Algebra

EXPONENTS

• Definition of Exponents: Exponents are a shorthand way of expressing repeated multiplication of a number by itself.
  • Example: $7 \times 7 \times 7 \times 7 \times 7 \times 7 = 7^6$ (read as "7 to the power 6")
• Basics:
  • If $a$ is the base and $n$ is the exponent, then:
  • $a^n = a \times a \times \ldots \times a \text{ (n times)}$
  • Special cases:
  • $a^0 = 1$ (for any non-zero $a$)
  • $a^{-n} = \frac{1}{a^n}$ (for any positive integer $n$)
  • Examples:
    • $5^0 = 1$,
    • $b \cdot b \cdot b \cdots = b^7$
    • $4^3 = 64$ and $\frac{1}{125} = 5^{-3}$
    • $2^3 = \sqrt[2]{\frac{1}{2^2}} = \sqrt{2}$

RULES OF EXPONENTS

• Basic Rules:
  1. Product Rule: $a^m \times a^n = a^{m+n}$
  2. Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$
  3. Power of a Power Rule: $(a^m)^n = a^{mn}$
  4. Power of a Product Rule: $(ab)^m = a^m \cdot b^m$
  5. Power of a Quotient Rule: $\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$

ORDER OF OPERATIONS

• To solve mathematical expressions, follow these steps:
  1. Evaluate expressions inside parentheses.
  2. Compute exponents and roots next.
  3. Perform multiplication and division from left to right.
  4. Lastly, perform addition and subtraction from left to right.

ALGEBRAIC EXPRESSIONS

• Definitions:
  • Variable: A symbol that represents an unknown value (e.g., $x$, $y$).
  • Term: A number or a product of numbers and variables (e.g., $3x$).
  • Expression: A combination of terms (e.g., $3x + 27y - 5$).
  • Constant Term: A term with a fixed value (e.g., -7 in $3xy + 12x - 7$).
  • Coefficient: The numerical factor multiplied by variables (e.g., 3 in $3xy$).
• Degree of Expression:
  • The degree is the highest exponent of any variable in the expression.
  • Examples:
  • $3x + 7$ (1st-degree)
  • $9x^2 - 4x + 1$ (2nd-degree)
  • $10y^3 + 4y^2 - y + 8$ (3rd-degree)

EQUATIONS AND FORMULAS

• Equations:
  • An equation asserts the equality of two expressions (indicated by $=$).
  • Solving the equation: Finding the value of the variable that makes the equation true.
  • Example: Solve $2x + 4 = 7x - 1$ to get $x = 1$.
• Simultaneous Equations: Set of equations that need to be solved together. Example:
  • $3x + 2y = 12$
  • $6x - y = 5$
• Formula: A mathematical representation using variables, e.g., perimeter of a rectangle $P = 2(L + W)$.
  • To find $P$ given $L$ and $W$, substitute their values.

SOLVING LINEAR EQUATIONS

• Steps:
  1. Move like terms to one side of the equation.
  2. Perform the same operations on both sides to isolate the variable (e.g., $7x + 3 = 2x + 18$ results in $x = 3$).

SIMULTANEOUS LINEAR EQUATIONS - ELIMINATION METHOD

• Steps:
  1. Adjust coefficients of one variable in both equations to match.
  2. Eliminate one variable by adding or subtracting equations.
  3. Solve the resulting equation for one variable.
  4. Substitute to find the second variable.
  5. Check solutions by substituting back into the original equations.

SIMULTANEOUS EQUATIONS - SUBSTITUTION METHOD

• Example: Solve equations such as:
  • $x + y = 7$
  • $3x - y = 8$
  • Steps:
  1. Express one variable in terms of the other (e.g., $x = 7 - y$).
  2. Substitute into the other equation to solve for the remaining variable.

ARITHMETIC AND GEOMETRIC SEQUENCES

• Definitions:
  • Arithmetic Sequence: Sequence where the difference between consecutive terms is constant.
  • Formula: $T_n = a + (n - 1)d$ where $a$ is the first term and $d$ is the common difference.
  • Geometric Sequence: Sequence where each term is found by multiplying the previous term by a constant.
  • Formula: $T_n = a \cdot r^{(n-1)}$ where $r$ is the common ratio.

APPLICATIONS

• Practical examples of equations in real life, such as determining costs or analyzing data patterns.
  • Example of identifying costs via linear equations: $x + 3x = 3000$ leads to identifying individual costs.