Session 01-03 - Core Algebra & Exponents

Section 01: Mathematical Foundations & Algebra

Author

Dr. Nikolai Heinrichs & Dr. Tobias Vlćek

Entry Quiz

Quick Review from Last Session

Complete individually, then we review the results together

  1. Express in set-builder notation: All odd integers less than 20
  2. If \(A = \{1, 3, 5, 7\}\) and \(B = \{3, 4, 5, 6\}\), find \(A \cap B\) and \(A \cup B\)
  3. Is \(0.\overline{36}\) rational? If yes, express as a fraction.
  4. True or false: If \(p \Rightarrow q\) is true and \(q\) is false, what can we say about \(p\)?

Ok, let’s review together!

Homework Discussion

Homework Showcase

15 minutes for presentations and discussion

  • Present and discuss your solutions from Tasks 01-02
  • Share any challenging problems or interesting approaches
  • This is your opportunity to ask questions and learn from each other

Remember: Explaining your solution helps solidify your understanding!

Learning Objectives

Learning Objectives

By the end of this session, you can:

  • Apply the order of operations (PEMDAS) reliably
  • Simplify expressions with the laws of exponents
  • Convert and compute in scientific notation
  • Solve absolute value equations and inequalities
  • Factor out common factors and recognize special patterns

Algebraic Expressions

“Algebraic”?

An algebraic expression combines:

  • Variables: letters representing unknown values (x, y, z, …)
  • Constants: fixed numbers (2, π, -5, …)
  • Operations: +, −, ×, ÷, and exponents
  • Example: \(x^3 - 2x^2 + 7\)

Not algebraic: \(\sin(x)\), \(\log(x)\), \(e^x\)

These are transcendental functions - but no need to worry about this for now!

Order of Operations - PEMDAS

  1. Parentheses (brackets, braces)
  2. Exponents (powers, roots)
  3. Multiplication and Division (left to right)
  4. Addition and Subtraction (left to right)

Example: \(2 + 3 \times 4^2 - (5 - 3) \div 2\)

  • Step 1 (Parentheses): \((5 - 3) = 2\)
  • Step 2 (Exponents): \(4^2 = 16\)
  • Steps 3-4: \(3 \times 16 = 48\), \(2 \div 2 = 1\), then \(2 + 48 - 1 = 49\)

Practice PEMDAS Together

Let’s evaluate step-by-step: \(\frac{3^2 + 2 \times (4 - 1)}{5 - 2}\)

  • Numerator first:
    • Parentheses: \((4 - 1) = 3\)
    • Exponent: \(3^2 = 9\)
    • Multiply: \(2 \times 3 = 6\)
    • Add: \(9 + 6 = 15\)
  • Denominator: \(5 - 2 = 3\)
  • Final division: \(\frac{15}{3} = 5\)

Individual Exercise 01

Work individually for 5 minutes

Evaluate and then we’ll review together:

  1. \(4 + 2^3 \times 3 - 12 \div 4\)
  2. \((3 + 2)^2 - 3 \times (7 - 4)\)
  3. \(\frac{2^3 + 3 \times 2}{10 - 3}\)
  4. \(5 \times [2 + 3 \times (4 - 2)^2]\)

Break

Laws of Exponents

Why Exponents Matter

Curves of y = x² and y = 2^x for x from 0 to 8. They cross at x = 2 and again at x = 4; beyond x = 4 the exponential 2^x rises far more steeply and stays above x².

Doubling (\(2^x\)) overtakes squaring (\(x^2\)) for good at \(x = 4\) - the engine behind compound interest, which we revisit today.

The Fundamental Rules

These laws help us manage exponents!

Rule Formula Example
Product Rule \(a^m \cdot a^n = a^{m+n}\) \(x^3 \cdot x^4 = x^7\)
Quotient Rule \(\frac{a^m}{a^n} = a^{m-n}\) \(\frac{x^5}{x^2} = x^3\)
Power Rule \((a^m)^n = a^{mn}\) \((x^3)^2 = x^6\)
Product Power \((ab)^n = a^n b^n\) \((2x)^3 = 8x^3\)
Quotient Power \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\) \(\left(\frac{x}{3}\right)^2 = \frac{x^2}{9}\)

Special Exponent Values

These are essential to memorize!

  • \(a^0 = 1\) (for any \(a \neq 0\))
  • \(a^1 = a\)
  • \(a^{-n} = \frac{1}{a^n}\)
  • \(a^{1/n} = \sqrt[n]{a}\)
  • \(a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m\)
WarningCommon Mistake

\((x + y)^2 \neq x^2 + y^2\), remember: \((x + y)^2 = x^2 + 2xy + y^2\)

More about this later!

Let’s Practice Together

Simplify: \(\frac{(3x^2y)^2 \cdot x^{-3}}{9xy^2}\)

  • Expand the power: \(\frac{9x^4y^2 \cdot x^{-3}}{9xy^2}\)
  • Combine exponents in numerator: \(\frac{9x^{4-3}y^2}{9xy^2}\)
  • Simplify: \(\frac{9x^1y^2}{9xy^2}\)
  • Finalize: \(\frac{1}{1} = 1\)

Your turn: Try \(\frac{(2a^3)^2 \cdot a^{-4}}{4a}\)

Pair Exercise 01

Work together in pairs for 5 minutes

Simplify completely:

  1. \((x^3)^2 \cdot x^{-4}\)
  2. \(\frac{12x^5y^3}{3x^2y}\)
  3. \(\left(\frac{2x^2}{y}\right)^3 \cdot \frac{y^2}{4x^3}\)
  4. \((3^2)^3 \cdot 3^{-5}\)

Scientific Notation

Why Scientific Notation?

Essential for extreme values!

Scientific notation: \(a \times 10^n\) where \(1 \leq |a| < 10\)

Real-world examples:

  • World population: \(8,000,000,000 = 8.0 \times 10^9\) people
  • Virus diameter: \(0.0000001 = 1 \times 10^{-7}\) meters
  • US National debt: \(\text{\$}31,400,000,000,000 = 3.14 \times 10^{13}\) dollars

Scientific notation makes calculations with very large or very small numbers practical!

Converting to Scientific Notation

Convert 56,700,000

  • Move decimal left to get one non-zero digit
  • Count moves: 7 positions left
  • Result: \(5.67 \times 10^7\)

Convert 0.00000423

  • Move decimal right to get one non-zero digit
  • Count moves: 6 positions right
  • Result: \(4.23 \times 10^{-6}\)
  • Moving decimal left → positive exponent
  • Moving decimal right → negative exponent

Operations with Scientific Notation

Multiplication: \((3 \times 10^5) \times (2 \times 10^3)\)

  • Multiply coefficients: \(3 \times 2 = 6\)
  • Add exponents: \(10^5 \times 10^3 = 10^8\)
  • Result: \(6 \times 10^8\)

Division: \(\frac{8.4 \times 10^7}{2.1 \times 10^4}\)

  • Divide coefficients: \(8.4 \div 2.1 = 4\)
  • Subtract exponents: \(10^7 \div 10^4 = 10^3\)
  • Result: \(4 \times 10^3\)

Coffee Break

Absolute Value

Understanding Absolute Value

The absolute value \(|x|\) represents the distance from zero

Definition:

\[|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}\]

Examples:

  • \(|5| = 5\), \(\quad|-3| = 3\), \(\quad|0| = 0\)

Think of absolute value as “removing the sign” or “distance without direction”

The Graph of \(|x|\)

V-shaped graph of y = |x| for x from -4 to 4, with its tip at the origin. The points (-3, 3) and (3, 3) are marked at the same height: -3 and 3 have the same absolute value, 3.

Both \(-3\) and \(3\) sit at the same height: they share the same distance from zero.

Properties of Absolute Value

These properties are fundamental for absolute values

  • Non-negativity: \(|x| \geq 0\) for all \(x\) (never negative)
  • Zero property: \(|x| = 0\) if and only if \(x = 0\)
  • Multiplicative: \(|xy| = |x| \cdot |y|\)
  • Quotient: \(\left|\frac{x}{y}\right| = \frac{|x|}{|y|}\) (when \(y \neq 0\))
  • Squares: \(|x|^2 = x^2\)
TipGeometric Interpretation

\(|x|\) as the distance from \(x\) to \(0\) on the number line.

Solving Absolute Value Equations

Example: Solve \(|2x - 6| = 4\)

Two cases to consider:

  • Case 1: \(2x - 6 = 4\)
    • \(2x = 10\)
    • \(x = 5\)
  • Case 2: \(2x - 6 = -4\)
    • \(2x = 2\)
    • \(x = 1\)
  • Solution: \(x \in \{1, 5\}\)

Working with Inequalities

Before we solve them, what are the rules?

An equation has solutions, an inequality has a solution set - usually a whole interval.

  • Add or subtract anything on both sides: sign stays
  • Multiply or divide by a positive number: sign stays
  • Multiply or divide by a negative number: sign flips

\(-2x < 6\) becomes \(x > -3\), not \(x < -3\).

Multiplying by \(-1\) mirrors the number line: \(2 < 3\), but \(-2 > -3\).

Inequality Chains

A chain like \(-2 < x - 3 < 2\) is two inequalities at once

  • Read it as: \(-2 < x - 3\) and \(x - 3 < 2\)
  • Whatever you do, do it to all three parts

Quick practice: Solve \(-1 \leq 3 - 2x < 5\) and give the answer in interval notation.

Absolute Value Inequalities

\(|x| < a\) means \(-a < x < a\)

Example: \(|x - 3| < 2\)

  • \(-2 < x - 3 < 2\)
  • \(1 < x < 5\)
  • Solution: \((1; 5)\)

\(|x| > a\) means \(x < -a\) OR \(x > a\)

Example: \(|x - 3| > 2\)

  • \(x - 3 < -2\) OR \(x - 3 > 2\)
  • \(x < 1\) OR \(x > 5\)
  • Solution: \((-\infty; 1) \cup (5; \infty)\)

Quality Control: Bolts must be \(20 \pm 0.3\) mm

  • Specification: \(|d - 20| \leq 0.3\)
  • Acceptable range: \([19.7; 20.3]\) mm

Solution Sets on the Number Line

The two inequality types look very different:

Two number lines with hollow dots at -2 and 2. For |x| < 2 a single segment between -2 and 2 is marked; for |x| > 2 two rays are marked, one to the left of -2 and one to the right of 2.

  • \(|x| < 2\): one interval around zero, \((-2; 2)\)
  • \(|x| > 2\): two rays pointing away from zero, \((-\infty; -2) \cup (2; \infty)\)

Basic Factorization

What Is Factoring For?

In product form you can see where an expression is zero

Factoring turns a sum into a product: \(12x^3 - 18x^2 + 6x = 6x(2x^2 - 3x + 1)\).

Zero-product rule: a product is zero exactly when one of its factors is zero.

\[A \cdot B = 0 \iff A = 0 \text{ or } B = 0\]

Example: \((x - 6)(x + 4) = 0 \iff x = 6\) or \(x = -4\)

Only with \(0\) on the other side: \(A \cdot B = 7\) does not mean \(A = 7\) or \(B = 7\).

What Does Zero Mean?

The zeros of a profit show where the profit starts and ends

Example: \(P = (x - 3)(11 - x)\) is a company’s profit, \(x\) the output in 100 units, \(P\) in 1,000 EUR. Only \(x > 0\) makes sense.

  • At \(x = 3\) the loss turns into a profit: the break-even point. At \(x = 11\) the profit turns back into a loss: the profit limit.
  • Same sign in both factors gives a positive product, different signs give a negative one
  • One test value fixes the sign between two zeros: \(x = 7\) gives \(16 > 0\)

Profit is positive exactly on \((3; 11)\).

Quick Check: Zero-Product Rule

Work individually for 3 minutes

  1. Solve \((x + 5)(x - 9) = 0\)

  2. Where is \((x - 4)(6 - x)\) positive?

Common Factor Method

Always check for common factors first!

Example: Factor \(12x^3 - 18x^2 + 6x\)

  • Find the GCF (Greatest Common Factor):
    • Numbers: GCF of 12, 18, 6 is 6
    • Variables: lowest power of x is \(x^1\)
    • GCF = \(6x\)
  • Factor out: \(6x(2x^2 - 3x + 1)\)

Difference of Squares

Pattern: \(a^2 - b^2 = (a + b)(a - b)\)

Examples:

  • \(x^2 - 9 = x^2 - 3^2 = (x + 3)(x - 3)\)
  • \(4x^2 - 25 = (2x)^2 - 5^2 = (2x + 5)(2x - 5)\)
  • \(16x^2 - 49y^2 = (4x)^2 - (7y)^2 = (4x + 7y)(4x - 7y)\)

\(a^2 + b^2\) cannot be factored! This is because we need a difference (subtraction) between two perfect squares to use this factorization pattern. When we have a sum (addition) like \(x^2 + 9\), there’s no real number factorization.

Perfect Square Trinomials

Recognize these special patterns

Pattern Formula How to Recognize
\((a + b)^2\) \(a^2 + 2ab + b^2\) Squares, middle = 2×(product of roots)
\((a - b)^2\) \(a^2 - 2ab + b^2\) Same, but middle term is negative

Examples:

  • \(x^2 + 6x + 9 = (x + 3)^2\)
  • \(x^2 - 10x + 25 = (x - 5)^2\)
  • \(4x^2 + 12x + 9 = (2x + 3)^2\)

Group Exercise 01

Work in groups for 8 minutes

Factor completely:

  1. \(3x^2 - 27\)
  2. \(x^2 - 8x + 16\)
  3. \(25x^2 - 9\)
  4. \(5x^3 - 20x\)
  5. \(x^2 + 14x + 49\)

Business Applications

Compound Interest Revisited

The power of exponential growth

Formula: \(A = P(1 + r)^t\)

  • P = Principal (initial amount)
  • r = Interest rate (as decimal), t = Time periods
  • A = Final amount

Example: 5,000 EUR at 6% annual interest

  • After 1 year: \(A = 5000(1.06)^1 = 5,300\text{ EUR}\)
  • After 10 years: \(A = 5000(1.06)^{10} = 8,954.24\text{ EUR}\)

Scientific Notation in Business

Data Analysis Example: A tech company processes:

  • Daily transactions: \(2.4 \times 10^8\)
  • Average transaction value: \(3.5 \times 10^1\) euros
  • Server cost per transaction: \(2 \times 10^{-4}\) euros

Calculate daily revenue and costs:

  • Revenue = \((2.4 \times 10^8) \times (3.5 \times 10^1)\) = \(8.4 \times 10^9\) euros
  • Costs = \((2.4 \times 10^8) \times (2 \times 10^{-4})\) = \(4.8 \times 10^4\) euros
  • Profit = 8.4 billion EUR - 48,000 EUR ≈ 8.4 billion EUR

Practice Integration

Individual Exercise 02

Work individually for 5 minutes

  1. Simplify: \(\frac{(3x^2)^3 \cdot x^{-5}}{9x^2}\)
  2. Factor: \(4x^2 - 36\)
  3. Solve: \(|3x - 9| = 6\)
  4. Express in scientific notation: The distance from Earth to Moon is 384,400 km
  5. Evaluate: \(2 + 3 \times 2^3 - 16 \div 4\)

Pair Exercise 02

Work in pairs for 5 minutes

A manufacturing company has:

  • Quality standard: Product weight must satisfy \(|w - 100| \leq 2\) grams
  • Daily production: \(3.2 \times 10^3\) unit
  1. What is the acceptable weight range?
  2. Express monthly production in scientific notation (25 working days)

Wrap-up

Key Takeaways

  • PEMDAS ensures consistent calculation order
  • Exponent laws are the foundation for all algebra
  • Scientific notation handles extreme values efficiently
  • Absolute value measures distance and defines tolerances
  • Basic factorization reveals structure in expressions

These are the foundation for all advanced mathematics!

Final Assessment

Quick Check

Work individually, then we discuss

  1. Simplify completely: \(\frac{(2x^3)^2 \cdot x^{-4}}{8x}\)
  2. A machine part passes quality control if its weight \(w\) (in grams) satisfies \(|w - 50| \leq 1.5\). State the acceptable weight range.

Next Session Preview

Coming Up

Advanced Factorization & Radicals

  • Factoring trinomials \(ax^2 + bx + c\)
  • Sum and difference of cubes
  • Roots, radicals, and rationalizing denominators
  • Complex algebraic manipulation

Complete Tasks 01-03!