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

Express in set-builder notation: All odd integers less than 20
If $A = \{1, 3, 5, 7\}$ and $B = \{3, 4, 5, 6\}$, find $A \cap B$ and $A \cup B$
Is $0.\overline{36}$ rational? If yes, express as a fraction.
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 - 15 Minutes

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

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

Parentheses (brackets, braces)
Exponents (powers, roots)
Multiplication and Division (left to right)
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 work through this step-by-step

Evaluate: $\frac{3^2 + 2 \times (4 - 1)}{5 - 2}$

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:

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

Break - 10 Minutes

Laws of Exponents

Why Exponents Matter

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 $a \neq 0$)
$a^1 = a$
$a^{-n} = \frac{1}{a^n}$
$a^{1/n} = \sqrt[n]{a}$
$a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$

. . .

Common Mistake

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

More about this later!

Calculator: Powers and Roots

Powers and roots can also be easily computed using a calculator:

. . .

Example 1: Calculate $5^3 + \sqrt[4]{16}$

. . .

Enter: 5 ^ 3 + SHIFT x^ 4 → 16 =

. . .

Result: 127

. . .

Example 2: Calculate $(2^3 + \sqrt[3]{27}) \times \sqrt[5]{32}$

. . .

Enter: ( 2 ^ 3 + SHIFT x^ 3 → 27 ) × SHIFT x^ 5 → 32 =

. . .

Result: 128

Calculator: Memory Variables

Store values for repeated use:

Variables available: A, B, C, D, E, F, x, y, M
Store: Enter value, press STO, then variable letter (no shift/alpha)
Recall: Press ALPHA + variable letter

. . .

Example: Store 3.14159 in A

Enter 3.14159 → STO → A
Use later: 2 × ALPHA A = gives 6.28318

. . .

Ans automatically stores your last result!

Calculator: The Ans Button

Using your last result efficiently:

The Ans button recalls your last calculation result.

Ans is automatically inserted when you start with an operator

. . .

Example: Calculate $5 + 3$, then multiply by 2:

. . .

Enter: 5 + 3 = → Result: 8
Enter: × 2 = → Calculator shows Ans × 2 = 16

. . .

Chain calculations become much faster with Ans!

Calculator: The CALC Button

Evaluate expressions with different values:

The CALC button lets you enter an expression with variables.

Enter an expression with variables (e.g., 3A + B)
Press CALC
Enter values when prompted and press = for the result

. . .

Example: Evaluate $3A + B$ for $A = 5$, $B = 10$

. . .

Enter: 3 ALPHA A + ALPHA B → CALC → 5 = 10 =
Result: 25

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:

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

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: $4 \times 10^3$

Calculator: Scientific Notation

For very large or small numbers:

Use the ×10ˣ button to enter scientific notation.

. . .

Example 1: Enter $6.022 \times 10^{23}$

. . .

Enter: 6.022 10ˣ 23 =

. . .

Example 2: Calculate $\frac{1.5 \times 10^8}{3 \times 10^4}$

. . .

Enter: 1.5 10ˣ 8 ÷ 3 10ˣ 4 =

. . .

Result: $5 \times 10^3 = 5000$

Coffee Break - 15 Minutes

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 $|x|$

. . .

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 working with absolute values

Non-negativity: $|x| \geq 0$ for all $x$ (absolute value is 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: $|x|^2 = x^2$

. . .

Geometric 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: $x \in \{1, 5\}$

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:

. . .

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

Basic Factorization

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
$(a + b)^2$	$a^2 + 2ab + b^2$	Perfect 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:

$3x^2 - 27$
$x^2 - 8x + 16$
$25x^2 - 9$
$5x^3 - 20x$
$x^2 + 14x + 49$

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 at 6% annual interest

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

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 - €48,000 ≈ €8.4 billion

Practice Integration

Individual Exercise 02

Work individually for 5 minutes

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

What is the acceptable weight range?
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

Simplify completely: $\frac{(2x^3)^2 \cdot x^{-4}}{8x}$
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!

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	\(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)	\(\left(\frac{x}{3}\right)^2 = \frac{x^2}{9}\)

Entry Quiz

Quick Review from Last Session

Homework Discussion - 15 Minutes

Homework Showcase

Learning Objectives

Learning Objectives

Algebraic Expressions

“Algebraic”?

Order of Operations - PEMDAS

Practice PEMDAS Together

Individual Exercise 01

Break - 10 Minutes

Laws of Exponents

Why Exponents Matter

The Fundamental Rules

Special Exponent Values

Calculator: Powers and Roots

Calculator: Memory Variables

Calculator: The Ans Button

Calculator: The CALC Button

Let’s Practice Together

Pair Exercise 01

Scientific Notation

Why Scientific Notation?

Converting to Scientific Notation

Operations with Scientific Notation

Calculator: Scientific Notation

Coffee Break - 15 Minutes

Absolute Value

Understanding Absolute Value

The Graph of \(|x|\)

Properties of Absolute Value

Solving Absolute Value Equations

Absolute Value Inequalities

Solution Sets on the Number Line

Basic Factorization

Common Factor Method

Difference of Squares

Perfect Square Trinomials

Group Exercise 01

Business Applications

Compound Interest Revisited

Scientific Notation in Business

Practice Integration

Individual Exercise 02

Pair Exercise 02

Wrap-up

Key Takeaways

Final Assessment

Quick Check

Next Session Preview

Coming Up