Session 01-06 - Synthesis & Applications

Section 01: Mathematical Foundations & Algebra

Author

Dr. Nikolai Heinrichs & Dr. Tobias Vlćek

Warm-up Quiz

Quick Skills Check

5 minutes - test your readiness

Simplify: $(2x^3)^2 \cdot x^{-5}$
Factor: $x^2 - 5x - 14$
Evaluate: $\log_2(32)$
Simplify: $\sqrt{48}$
Express in scientific notation: $0.000425$

. . .

These cover the core skills - let’s review together!

Solution Presentations

Showcase Your Solutions

25 minutes for comprehensive discussions

Present your most challenging problem from Tasks 01-05
Are there any tasks from previous lectures you want to discuss?
Explain your problem-solving strategy
Share any connections between topics you discovered

. . .

Today we see how all pieces fit together!

Key Concepts

Set Theory & Number Systems

Essential foundations

Number hierarchy: $ $\mathbb{I}$ $

Set operations:

Union: $A \cup B$ (all elements in either)
Intersection: $A \cap B$ (elements in both)
Difference: $A \setminus B$ (in A but not B)

Quick practice:

If $A = \{1, 3, 5\}$ and $B = \{3, 4, 5, 6\}$
Find $A \cap B$ and $A \cup B$

Exponent Laws

Your main tools for exponents

Rule	Formula	Quick Example
Product	$a^m \cdot a^n = a^{m+n}$	$x^3 \cdot x^4 = x^7$
Quotient	$\frac{a^m}{a^n} = a^{m-n}$	$\frac{x^5}{x^2} = x^3$
Power	$(a^m)^n = a^{mn}$	$(x^3)^2 = x^6$
Negative	$a^{-n} = \frac{1}{a^n}$	$x^{-2} = \frac{1}{x^2}$

. . .

Quick practice: Simplify $\frac{(3x^2)^3}{9x^4}$

Factorization Techniques

Your factoring toolbox

Common factor: Always check first!
Difference of squares: $a^2 - b^2 = (a+b)(a-b)$
Perfect squares: $a^2 \pm 2ab + b^2 = (a \pm b)^2$
AC method: For $ax^2 + bx + c$ when $a \neq 1$
Grouping: For 4-term polynomials
Cubes: $a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)$
Substitution: Let $u = $ common expression

. . .

Quick practice: Factor $x^4 - 16$

Radicals & Logarithms

Advanced operations

Radicals:

Simplify by extracting perfect powers
Rationalize using conjugates: $(a + \sqrt{b})(a - \sqrt{b}) = a^2 - b$

Logarithms:

$\log_a(xy) = \log_a(x) + \log_a(y)$
$\log_a(x^n) = n\log_a(x)$
If $a^x = b$, then $x = \log_a(b)$

Quick practice: Solve $2^x = 10$

Practice Block 1 - Fundamentals

Individual Exercise

10 minutes - repeat the basics

Simplify: $\frac{x^3 \cdot x^{-5}}{x^{-3}}$
Factor completely: $3x^2 - 27$
Rationalize: $\frac{6}{\sqrt{3}}$
Solve: $|3x - 6| = 9$
Evaluate: $\log_3(81) - \log_3(3)$
Express in scientific notation: 45,600,000

Break - 10 Minutes

Practice Block 2 - Fundamentals

Pair Exercise

15 minutes - work together

Factor using substitution: $x^4 - 5x^2 + 4$
Simplify: $\sqrt{75} + \sqrt{48} - \sqrt{27}$
If $2^x = 3$ and $3^y = 5$, find $2^{xy}$
Rationalize: $\frac{4}{3 - \sqrt{5}}$
Factor by grouping: $x^3 + 2x^2 - 9x - 18$

Mixed Techniques

Mixed Technique Problems

Combine multiple methods

Factor: $8x^3 - 27y^3$
Simplify: $\frac{x^3 - 8}{x^2 - 4} \cdot \frac{x + 2}{x^2 + 2x + 4}$
If $|2x - 5| < 7$, find the solution interval
Rationalize: $\frac{2}{\sqrt{6} - \sqrt{2}}$

Business Applications

Financial Growth Problems

Real-world application with compound interest

An investment of €5,000 earns 8% annual interest compounded yearly.

Write the formula for the amount after $t$ years
Calculate the value after 10 years
When will it triple in value?

Cost-Profit Analysis

Combining algebraic techniques for Manufacturing

A company produces $x$ units with:

Cost: $C = 2x^2 + 100x + 5000$
Revenue: $R = 500x - 3x^2$

Your tasks:

How can you compute the profit?
Factor the profit completely

Coffee Break - 15 Minutes

Another Block

Synthesis Problems

Comprehensive practice

Solve: $\log_3(x^2 - 8) = 2$
Factor: $x^3 + 3x^2 - x - 3$
Solve: $\log_2(x + 5) = 3$
Factor: $x^3 - 27$
Simplify: $\frac{2\sqrt{12} + 3\sqrt{27}}{\sqrt{3}}$

Challenge Problems

Test your skills by solving these in pairs

If $3^x + 3^{-x} = 4$, find $9^x + 9^{-x}$
Factor completely: $x^6 - 7x^3 - 8$
A bacteria culture triples every 4 hours. Starting with 500 bacteria:
- When will there be 1 million bacteria?
- Express the population after 1 day in scientific notation

Wrap-up

Key Skills Mastery

Your Section 1 foundation is complete!

Can you factor any polynomial type we’ve studied?
Can you apply all exponent laws confidently?
Can you simplify complex radical expressions?
Can you solve logarithmic equations?
Can you work with absolute values?
Can you convert to/from scientific notation?
Most importantly: Can you combine techniques to solve complex problems?

Study Strategy Going Forward

Success in mathematics requires consistent practice

Daily (10-15 minutes):

Review one concept from past sections
Solve 2-3 smaller problems
Check your solutions carefully

. . .

Do this in addition to our lecture and the problems we solve together here.

Moving Forward

You’ve built a strong foundation!

Your achievements:

Mastered algebraic foundations
Learned powerful problem-solving techniques
Built confidence with complex problems
Prepared for advanced mathematics

Next: Section 2 - Functions and Calculus Basics

. . .

Keep practicing Section 1 skills alongside new material - they remain essential !

Questions & Reflection

Open Discussion

Share your thoughts

What topic needs more review?
Which connections were most helpful?
Any remaining questions?
How do you feel about the material?

. . .

Mathematics builds on itself - your foundation is now solid!

Rule	Formula	Quick Example
Product	\(a^m \cdot a^n = a^{m+n}\)	\(x^3 \cdot x^4 = x^7\)
Quotient	\(\frac{a^m}{a^n} = a^{m-n}\)	\(\frac{x^5}{x^2} = x^3\)
Power	\((a^m)^n = a^{mn}\)	\((x^3)^2 = x^6\)
Negative	\(a^{-n} = \frac{1}{a^n}\)	\(x^{-2} = \frac{1}{x^2}\)