Power Lab Answer Key 32-PT Power Lab Answer Key
Master Solution Key: 32-PT Protocol
Class Code ALG2-EXP-32
I
Phase I: Conversions
01. \( 121^{\frac{1}{2}} \)
02. \( 27^{\frac{1}{3}} \)
03. \( x^{\frac{5}{4}} \)
04. \( y^{\frac{2}{5}} \)
05. \( \sqrt{36} \)
06. \( \sqrt[3]{125} \)
07. \( \sqrt[4]{z^3} \)
08. \( \sqrt[5]{w^4} \)
II
Phase II: Unit Fractions
09. 7 \( \sqrt{49} \)
10. 10 \( \sqrt[3]{1000} \)
11. 3 \( \sqrt[4]{81} \)
12. 2 \( \sqrt[5]{32} \)
13. 12 \( \sqrt{144} \)
14. 3 \( \sqrt[5]{243} \)
The Reactor Core (27-32)
27. \( 4^1 \) 4
28. \( 3^3 \) 27
29. \( 100^1 \) 100
30. \( 4^{-3} \) \( \frac{1}{64} \)
31. \( \sqrt{9} \cdot 1 \) 3
32. \( 8^{\frac{2}{3}} \cdot \frac{1}{4} = 4 \cdot \frac{1}{4} \) 1
III
Phase III: Rational Powers
15. \( (\sqrt[3]{8})^2 = 2^2 \) 4
16. \( (\sqrt{16})^3 = 4^3 \) 64
17. \( (\sqrt[5]{32})^2 = 2^2 \) 4
18. \( (\sqrt[3]{64})^2 = 4^2 \) 16
19. \( (\sqrt[3]{125})^2 = 5^2 \) 25
20. \( (\sqrt[4]{81})^3 = 3^3 \) 27
IV
Phase IV: Negatives & Bases
21. \( \frac{1}{\sqrt{25}} \) \( \frac{1}{5} \)
22. \( \frac{1}{(\sqrt[3]{8})^2} \) \( \frac{1}{4} \)
23. \( \frac{1}{(\sqrt[4]{16})^3} \) \( \frac{1}{8} \)
24. \( \frac{\sqrt{4}}{\sqrt{9}} \) \( \frac{2}{3} \)
25. \( \left(\frac{\sqrt[3]{8}}{\sqrt[3]{27}}\right)^2 \) \( \frac{4}{9} \)
26. \( 64^{\frac{1}{3}} = 4 \) 4
Confidential: Instructor Use Only
Power Lab Teacher Guide Teacher Facilitation Guide
Operation: Power Lab Rational Exponents
Lesson Code ALG2-7G-EXP
Lesson Mission
Students will transition from understanding radical expressions to manipulating them as rational exponents. The "Power Lab" theme frames the learning as a scientific investigation, where accuracy is critical for "core stability." This lesson moves from basic structural recognition to multi-step numeric simplification.
Primary TEKS
2A.7G: Rewrite radical expressions to equivalent forms using rational exponents.
Prerequisite
Perfect squares (1-15), perfect cubes (1-5), and basic exponent rules.
Inventory
Power Lab Slide Deck
Student Research Log (WS)
Master Solution Key
Scientific Calculator (Opt)
Deployment Schedule (50-60 min)
0-10m
Pre-Analysis (Hook)
Use Slide 3 (Anatomy) to discuss why we might prefer exponents over radical signs (formatting, operations, derivatives later in Calculus).
10-25m
Direct Instruction
Walk through Slides 4-7. Focus on the "Root First" strategy for Phase III. It is crucial students understand that roots make numbers smaller and easier to manage mentally.
25-50m
Active Experimentation
Students work through the Power Lab Worksheet. Monitor Phase III and IV carefully—these are where common calculation errors occur.
50-60m
Core Stabilization (Exit)
Review Problem 16 or 17 as a whole class. Have students "self-diagnose" their accuracy levels using the Reactor Core stability metaphor.
Critical Malfunctions (Common Errors)
The "Numerator Trap"
Students often swap the power and the index. Remind them: Index is the Denominator (Down-under) .
The "Negative Number" Myth
Students may think a negative exponent makes the value negative. Use the phrase: "Negative power = Reciprocal (the flip), NOT negative result."
Power First, Root Last
Squaring 27 is hard (729). Taking the cube root of 27 is easy (3). Always emphasize taking the Root first to keep numbers small.
The Zero Index
Students might confuse \( x^{1/n} \) with \( x^1 \). Remind them that a fraction as an exponent always implies a root operation.
Laboratory Modifications
For Struggling Researchers
Provide a "Perfect Power Cheat Sheet" (listing squares up to 225, cubes up to 125, and powers of 2 up to 2^6). Allow them to highlight the denominators (roots) in Phase I before calculating.
For Lead Scientists (Adv.)
Challenge them to create their own "Reactor Core" problem that involves at least three different exponent properties. Have them swap with a partner to solve.
© 2026 Power Lab Educational Systems Internal Document: TR-X99
Power Lab Worksheet Reactor Focus v3-Final-V2-MaxSpace POWER LAB
Rational Exponent Experiment Log
Name
Date
I
Phase I: Form Conversion
Convert the following expressions from radical form to exponential form, or vice versa.
01. \( \sqrt{121} \)
02. \( \sqrt[3]{27} \)
03. \( \sqrt[4]{x^5} \)
04. \( \sqrt[5]{y^2} \)
05. \( 36^{\frac{1}{2}} \)
06. \( 125^{\frac{1}{3}} \)
07. \( z^{\frac{3}{4}} \)
08. \( w^{\frac{4}{5}} \)
II
Phase II: Unit Fractions
Evaluate the following expressions by finding the indicated principal root.
09. \( 49^{\frac{1}{2}} \)
10. \( 1000^{\frac{1}{3}} \)
11. \( 81^{\frac{1}{4}} \)
12. \( 32^{\frac{1}{5}} \)
13. \( 144^{\frac{1}{2}} \)
14. \( 243^{\frac{1}{5}} \)
III
Phase III: Rational Powers
Simplify the expressions using the property \( a^{\frac{m}{n}} = (\sqrt[n]{a})^m \).
15. \( 8^{\frac{2}{3}} \)
16. \( 16^{\frac{3}{2}} \)
17. \( 32^{\frac{2}{5}} \)
18. \( 64^{\frac{2}{3}} \)
19. \( 125^{\frac{2}{3}} \)
20. \( 81^{\frac{3}{4}} \)
IV
Phase IV: Negatives & Bases
Apply reciprocal and quotient properties to simplify negative exponents or fractional bases.
21. \( 25^{-\frac{1}{2}} \)
22. \( 8^{-\frac{2}{3}} \)
23. \( 16^{-\frac{3}{4}} \)
24. \( \left(\frac{4}{9}\right)^{\frac{1}{2}} \)
25. \( \left(\frac{8}{27}\right)^{\frac{2}{3}} \)
26. \( \left(\frac{1}{64}\right)^{-\frac{1}{3}} \)
The Reactor Core
Critical Difficulty
Final Phase: System Synthesis
Simplify to a single value. Provide all intermediate steps in the laboratory space below.
Specimen 27 Evaluate: \( 4^{\frac{1}{2}} \cdot 4^{\frac{1}{2}} \)
Specimen 28 Evaluate: \( \left(9^{\frac{1}{2}}\right)^3 \)
Specimen 29 Evaluate: \( 100^{\frac{3}{2}} \div 100^{\frac{1}{2}} \)
Specimen 30 Evaluate: \( \left(16^{\frac{1}{2}}\right)^{-3} \)
Specimen 31 Evaluate: \( \sqrt{81^{\frac{1}{2}}} \cdot 5^0 \)
Specimen 32 Evaluate: \( \left(\frac{1}{8}\right)^{-\frac{2}{3}} \cdot 4^{-1} \)
Power Lab Simulation v5.4 Experimental Log: 32-PT Subject: Rational Exponents
Power Lab Slides Upgrade 3-FINAL-POLISH The Power Lab
Simplifying Rational Exponents
Algebra II
TEKS 2A.7G
Mission Parameters
Convert between radical form and rational exponent form.
Evaluate numeric expressions with unit fractions.
Simplify complex powers with negative exponents.
Synthesize exponent rules to solve multi-step expressions.
Subject Anatomy
a
Base
m
Power
Numerator
n
Root
Index (Denominator)
\( a^{\frac{m}{n}} = \sqrt[n]{a^m} \)
\( a^{\frac{m}{n}} = (\sqrt[n]{a})^m \)
I
Phase I: Conversion Protocol
Example A
\( \sqrt[3]{x^5} \rightarrow x^{5/3} \)
Example B
\( y^{2/7} \rightarrow \sqrt[7]{y^2} \)
Researcher Tip
The denominator is always the index (the root).
Think: Roots are under the ground, so the root is the under number.
II
Phase II: Unit Fractions
Level 1
\( 25^{1/2} \)
\( \sqrt{25} = 5 \)
Level 2
\( 64^{1/3} \)
\( \sqrt[3]{64} = 4 \)
Level 3
\( 16^{1/4} \)
\( \sqrt[4]{16} = 2 \)
Resulting in base integer values
III
Phase III: The Two-Step Process
1 Take the ROOT first
2 RAISE to the power
Why? Small numbers are easier!
SOLVE THIS
\( 27^{2/3} \)
\( (\sqrt[3]{27})^2 \)
\( (3)^2 \)
\( 9 \)
IV
Phase IV: Negative Charge
Negative exponents FLIP the expression to the other side of the fraction bar.
\( x^{-m/n} = \frac{1}{x^{m/n}} \)
Lab Simulation
\( 16^{-1/2} \) \( \frac{1}{4} \)
\( 8^{-1/3} \) \( \frac{1}{2} \)
\( 4^{-3/2} \) \( \frac{1}{8} \)
The Reactor Core
Critical Challenge
\( (25^{1/2} \cdot 25^{1/2})^{-1} \)
Multiply Bases Negative Flip
ANSWER: 1/25
Experiment Concluded
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