Module 01: Multi-Base Systems

Base Builders

Mastering the Product of Powers Property with Rational Exponents

The Assembly Puzzle

You are building a component where the length is \(x^{1/2}\) and the width is \(x^{1/3}\).

"How do we combine these different measurements into one expression?"

\(x^{1/2} \cdot x^{1/3} = ?\)

The Blueprint Rule

Product of Powers Property

\(x^a \cdot x^b = x^{a+b}\)

"Multiplying like bases? Add the blueprints (exponents) together."

Fractional Foundation

To add exponents with different denominators:

1

Find a Common Denominator

2

Convert the Fractions

3

Add the Numerators

Blueprint Walkthrough

\(x^{1/2} \cdot x^{1/3}\)

→ Identify the common denominator: 6

→ Convert: \( \frac{1}{2} = \frac{3}{6} \) and \( \frac{1}{3} = \frac{2}{6} \)

→ Add: \( \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \)

Final Specification

\(x^{5/6}\)

Structural Practice

Level: Apprentice

\(y^{3/4} \cdot y^{1/8}\)

Level: Engineer

\(z^{2/3} \cdot z^{1/2}\)

Complex Assemblies

Remember: You can only combine exponents if the bases are identical.

Danger Zone: Don't combine \(x\) and \(y\)!

Solve this structure:

\((x^{1/2} y^{2/3}) \cdot (x^{1/4} y^{1/6})\)

Base x: \(1/2 + 1/4 = ?\)

Base y: \(2/3 + 1/6 = ?\)

Inspection Complete

You are now ready to begin the **Base Builders** worksheet.

Target 1

Add fractional exponents with common denominators.

Target 2

Apply common denominators to unlike exponents.

Exponent Engineering

Base Builders Multiplying Bases

Fraction Friction Dividing Bases

Power Surge Raising Powers

Expansion Blueprint Distribution Rules

The Final Blueprint Station Rotation

Base Builders Worksheet

Base Builders Slides

Fraction Friction Worksheet

Fraction Friction Slides

Power Surge Worksheet

Power Surge Slides

Expansion Blueprint Worksheet

Expansion Blueprint Slides

Station Rotation Cards

Station Rotation Recording Sheet

Similar Sequences

Function Frontiers

Jessica Waters

Urban Concurrency Planning

Power Root Mastery

Power of Growth

Pattern Detectives

Recursive Sequence Mastery

Exponent Mastery

Slope and Symmetry

Algebra Rescue Polynomials

Domain Restriction Diagnostics

Optimal Apex Optimization

Slope Mastery

Polynomial Powers

Symmetry Secrets

Constant Quest Algebra Intervention