Section 01: Mathematical Foundations & Algebra
Complete individually, then we review the results together
Ok, let’s review together!
20 minutes for presentations and discussion
Remember: Explaining your solution helps solidify your understanding!
An algebraic expression combines:
Not algebraic: \(\sin(x)\), \(\log(x)\), \(e^x\)
These are transcendental functions - but no need to worry about this for now!
Example: \(2 + 3 \times 4^2 - (5 - 3) \div 2\)
Let’s work through this step-by-step
Evaluate: \(\frac{3^2 + 2 \times (4 - 1)}{5 - 2}\)
Practice order of operations for yourself
Evaluate and then we’ll review together:
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 | \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\) | \(\left(\frac{x}{3}\right)^2 = \frac{x^2}{9}\) |
These are essential to memorize!
Common Mistake
\((x + y)^2 \neq x^2 + y^2\), remember: \((x + y)^2 = x^2 + 2xy + y^2\)
More about this later!
Simplify: \(\frac{(3x^2y)^2 \cdot x^{-3}}{9xy^2}\)
Your turn: Try \(\frac{(2a^3)^2 \cdot a^{-4}}{4a}\)
Work together on these exponent problems in pairs
Simplify completely:
Essential for extreme values!
Scientific notation: \(a \times 10^n\) where \(1 \leq |a| < 10\)
Real-world examples:
Scientific notation makes calculations with very large or very small numbers practical!
Convert 56,700,000
Convert 0.00000423
Multiplication: \((3 \times 10^5) \times (2 \times 10^3)\)
Division: \(\frac{8.4 \times 10^7}{2.1 \times 10^4}\)
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:
Think of absolute value as “removing the sign” or “distance without direction”
These properties are fundamental for working with absolute values
Geometric Interpretation
\(|x|\) as the distance from \(x\) to \(0\) on the number line.
Example: Solve \(|2x - 6| = 4\)
Two cases to consider:
\(|x| < a\) means \(-a < x < a\)
Example: \(|x - 3| < 2\)
\(|x| > a\) means \(x < -a\) OR \(x > a\)
Example: \(|x - 3| > 2\)
Quality Control: Bolts must be \(20 \pm 0.3\) mm
Always check for common factors first!
Example: Factor \(12x^3 - 18x^2 + 6x\)
Pattern: \(a^2 - b^2 = (a + b)(a - b)\)
Examples:
\(a^2 + b^2\) 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.
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:
Practice factorization together (8 minutes)
Factor completely:
The power of exponential growth
Formula: \(A = P(1 + r)^t\)
Example: €5,000 at 6% annual interest
Data Analysis Example:
A tech company processes:
Calculate daily revenue and costs:
Apply your new skills individually
Business application problem
A manufacturing company has:
These are the foundation for all advanced mathematics!
Homework: Complete Tasks 01-03
Preview of Session 01-04:
Master these fundamentals - they’re the building blocks for everything else!
Your questions and insights are welcome!
Keep practicing - algebra gets easier with repetition!
Session 01-03 - Core Algebra & Exponents | Dr. Nikolai Heinrichs & Dr. Tobias Vlćek | Home