Power Towers Worksheet
Algebra Explorers
Power Towers
Unit: Exponents
Power of a Power Rule
Name:
Date:
Class:
The Secret Code: Discovery
What happens when we raise an exponent to another power? Let's unpack the code by writing out the expression in full.
The Power Tower
\( (x^3)^2 \)
Write Out the Square
\( (x^3) \cdot (x^3) \)
Unpack & Simplify
\( x \cdot x \cdot x \cdot x \cdot x \cdot x = x^6 \)
The Blueprint Rule
Formula
\( (x^a)^b = x^{a \cdot b} \)
Instead of expanding every time, use the shortcut: Multiply the exponents together! Keep the base exactly the same.
Scaffolded Construction: Guided Walkthroughs
Case 1: Variable Base
Simplify \( (y^5)^3 \)
1
Identify base & powers:
Base is \( y \); powers are \( 5 \) and \( 3 \).
2
Write as multiplication:
\( y^{5 \cdot 3} \)
3
Simplify:
\( y^{15} \)
Case 2: Numerical Base
Simplify \( (2^3)^2 \)
1
Multiply the exponents:
\( 2^{3 \cdot 2} = 2^6 \)
2
Expand the value:
Calculate \( 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \)
3
Final Standard Value:
\( 64 \)
Page 1 of 2
Power Towers Worksite
Leveling Up Practice
Phase 1: Guided Practice (Fill-in-the-blanks)
Problem A Variable
Simplify \( (x^4)^3 = \) \( x \) \( 4 \times \_\_ \) \( = x \) \( \_\_ \)
Problem B Numerical
Simplify \( (3^2)^3 = 3 \) \( 2 \times \_\_ \) \( = 3 \) \( \_\_ \) \( = \) \( \_\_\_\_\_ \)
Problem C Variable
Simplify \( (a^2)^5 = a \) \( 2 \times \_\_ \) \( = a \) \( \_\_ \)
Problem D Numerical
Simplify \( (5^1)^3 = 5 \) \( 1 \times \_\_ \) \( = 5 \) \( \_\_ \) \( = \) \( \_\_\_\_\_ \)
Phase 2: Independent Practice
Simplify fully
- Variable Base
\( (x^6)^4 \)
- Numerical Base
\( (3^2)^2 \)
- Variable Base
\( (m^3)^7 \)
- Numerical Base
\( (10^3)^2 \)
- Variable Base
\( (p^8)^3 \)
- Numerical Base
\( (4^1)^2 \)
The Penthouse Challenge
Can we apply the shortcut rule to a double tower? Multiply your way up to find the single simplified expression:
\( \Big( (x^2)^3 \Big)^2 \)
Workspace (Show steps!)
Answer: \( x \)
Page 2 of 2
Power Towers Answer Key
Teacher Edition
Power Towers Answer Key
Unit: Exponents
Power of a Power Rule
Name: TEACHER COPY
Date: TODAY
Class: ALGEBRA 1
The Secret Code: Discovery
What happens when we raise an exponent to another power? Let's unpack the code by writing out the expression in full.
The Power Tower
\( (x^3)^2 \)
Write Out the Square
\( (x^3) \cdot (x^3) \)
Unpack & Simplify
\( x \cdot x \cdot x \cdot x \cdot x \cdot x = x^6 \)
The Blueprint Rule
Formula
\( (x^a)^b = x^{a \cdot b} \)
Instead of expanding every time, use the shortcut: Multiply the exponents together! Keep the base exactly the same.
Scaffolded Construction: Guided Walkthroughs
Case 1: Variable Base
Simplify \( (y^5)^3 \)
1
Identify base & powers:
Base is \( y \); powers are \( 5 \) and \( 3 \).
2
Write as multiplication:
\( y^{5 \cdot 3} \)
3
Simplify:
\( y^{15} \)
Case 2: Numerical Base
Simplify \( (2^3)^2 \)
1
Multiply the exponents:
\( 2^{3 \cdot 2} = 2^6 \)
2
Expand the value:
Calculate \( 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \)
3
Final Standard Value:
\( 64 \)
Page 1 of 2 (Key)
Power Towers Worksite (Answer Key)
Leveling Up Practice
Phase 1: Guided Practice (Fill-in-the-blanks)
Problem A Variable
Simplify \( (x^4)^3 = \) \( x \) \( 4 \times 3 \) \( = x \) \( 12 \)
Problem B Numerical
Simplify \( (3^2)^3 = 3 \) \( 2 \times 3 \) \( = 3 \) \( 6 \) \( = \) \( 729 \)
Problem C Variable
Simplify \( (a^2)^5 = a \) \( 2 \times 5 \) \( = a \) \( 10 \)
Problem D Numerical
Simplify \( (5^1)^3 = 5 \) \( 1 \times 3 \) \( = 5 \) \( 3 \) \( = \) \( 125 \)
Phase 2: Independent Practice
Simplify fully
- Variable Base
\( (x^6)^4 \)
\( = x^{6 \cdot 4} = x^{24} \)
- Numerical Base
\( (3^2)^2 \)
\( = 3^{2 \cdot 2} = 3^4 = 81 \)
