Exponent Laws Interactive Foldable
Laws of Exponents Foldable
7th Grade Math • Guided Discovery & Notebook Organizer
Name:
Period:
Tab 1: Anatomy of a Power & Exponential Notation Foundation
Parts of a Power
5 3 = Power
5 = Base 3 = Exponent
• Base: The number being repeatedly multiplied as a factor.
• Exponent: Indicates how many ____________ to multiply the base by itself.
Example: \( 4^3 \) → Expanded: \( 4 \cdot 4 \cdot 4 \) → Value: _________
Tab 2 • Product Rule
Proof by Expansion:
\( 2^3 \cdot 2^4 = (2 \cdot 2 \cdot 2) \cdot (2 \cdot 2 \cdot 2 \cdot 2) = 2^{\underline{\hspace{16px}}} \)
Count the total factors: \( 3 + 4 = 7 \) factors of 2.
Rule: When multiplying powers with the same base, ____________ the exponents.
\( \text{Base}^m \cdot \text{Base}^n = \text{Base}^{\underline{\hspace{28px}}} \)
Worked Model:
\( 3^2 \cdot 3^5 = 3^{2+5} = 3^7 \)
Your Turn • Write in exponential form:
- \( 7^3 \cdot 7^8 \)
= ____________
- \( 2^4 \cdot 2^5 \)
= ____________
Tab 3 • Quotient Rule
Proof by Expansion:
\( \frac{3^5}{3^2} = \frac{3 \cdot 3 \cdot 3 \cdot 3 \cdot 3}{3 \cdot 3} = 3^{\underline{\hspace{16px}}} \)
Cancel common factors: \( 5 - 2 = 3 \) factors remain.
Rule: When dividing powers with the same base, ____________ the exponents.
\( \frac{\text{Base}^m}{\text{Base}^n} = \text{Base}^{\underline{\hspace{28px}}} \)
Worked Model:
\( \frac{5^7}{5^3} = 5^{7-3} = 5^4 \)
Your Turn • Write in exponential form:
- \( \frac{9^9}{9^4} \)
= ____________
- \( \frac{4^8}{4^2} \)
= ____________
Tab 4 • Zero Exponent Rule
Proof by Quotient Rule & Pattern:
\( \frac{5^3}{5^3} = 5^{3-3} = 5^0 \) Also \( \frac{125}{125} = 1 \)
\( 2^3=8 \rightarrow 2^2=4 \) \( 2^1=2 \xrightarrow{\div 2} 2^0 = \underline{\hspace{12px}} \)
Rule: Any non-zero base raised to power 0 equals _______.
\( \text{Base}^0 = \underline{\hspace{20px}} \quad (\text{Base} \neq 0) \)
Worked Model (Watch Parentheses!):
\( (6 \cdot 3)^0 = 1 \quad \text{but} \quad 6 \cdot 3^0 = 6(1) = 6 \)
Your Turn • Evaluate:
- \( (-14)^0 + 8^0 \)
= ____________
- \( 5 \cdot 4^0 - (5 \cdot 4)^0 \)
= ____________
Tab 5 • Mixed Practice
Multi-Step Strategy:
1. Multiply numerators (Product Rule).
2. Divide matching bases (Quotient Rule).
3. Evaluate any \( \text{Base}^0 = 1 \).
Multi-Step Model:
\( \frac{2^5 \cdot 2^3}{2^4} = \frac{2^{5+3}}{2^4} = \frac{2^8}{2^4} = 2^{8-4} = 2^4 = 16 \)
Challenge • Simplify completely:
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\( \frac{3^7 \cdot 3^3}{3^6} \) Show steps
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\( \frac{5^4 \cdot 5^3}{5^7} \) Show steps
Foldable Directions: Cut along outer solid border • Fold along dashed center lines • Glue spine into math notebook.
Page 1 • Student Copy
Exponent Laws Foldable Key
Laws of Exponents Foldable
Teacher Key
7th Grade Math • Master Reference & Misconception Guide
Complete Solutions
Tab 1: Anatomy of a Power & Exponential Notation Completed Notes
Parts of a Power
5 3 = Power
5 = Base 3 = Exponent
• Base: The number being repeatedly multiplied as a factor.
• Exponent: Indicates how many times / factors to multiply the base by itself.
Example: \( 4^3 \) → Expanded: \( 4 \cdot 4 \cdot 4 \) → Value: 64
Tab 2 • Product Rule Key
Discovery Expansion:
\( 2^3 \cdot 2^4 = (2 \cdot 2 \cdot 2) \cdot (2 \cdot 2 \cdot 2 \cdot 2) = 2^{\mathbf{\textcolor{#15803d}{7}}} \)
3 factors + 4 factors = 7 total factors of 2.
Rule: When multiplying powers with the same base, ADD the exponents.
\( \text{Base}^m \cdot \text{Base}^n = \text{Base}^{\mathbf{\textcolor{#15803d}{m+n}}} \)
Teacher Tip (Base Stays the Same):
\( 3^2 \cdot 3^5 = 3^{2+5} = 3^7 \) (Keep base 3, do not multiply \(3 \cdot 3 = 9\))
Answer Key • Solutions:
- \( 7^3 \cdot 7^8 \)
= \( 7^{3+8} = \mathbf{7^{11}} \)
- \( 2^4 \cdot 2^5 \)
= \( 2^{4+5} = \mathbf{2^9} \)
Tab 3 • Quotient Rule Key
Discovery Expansion:
\( \frac{3^5}{3^2} = \frac{3 \cdot 3 \cdot 3 \cdot \cancel{3} \cdot \cancel{3}}{\cancel{3} \cdot \cancel{3}} = 3^{\mathbf{\textcolor{#15803d}{3}}} \)
2 common factors of 3 cancel out; 3 remain.
Rule: When dividing powers with the same base, SUBTRACT the exponents.
\( \frac{\text{Base}^m}{\text{Base}^n} = \text{Base}^{\mathbf{\textcolor{#15803d}{m-n}}} \)
Teacher Tip (Base Stays the Same):
\( \frac{5^7}{5^3} = 5^{7-3} = 5^4 \) (Keep base 5, do not divide \(5 \div 5 = 1\))
Answer Key • Solutions:
- \( \frac{9^9}{9^4} \)
= \( 9^{9-4} = \mathbf{9^5} \)
- \( \frac{4^8}{4^2} \)
= \( 4^{8-2} = \mathbf{4^6} \)
Tab 4 • Zero Exponent Key
Discovery Proof Solutions:
\( \frac{5^3}{5^3} = 5^{3-3} = 5^0 \) \( \frac{125}{125} = \mathbf{1} \implies 5^0 = 1 \)
Pattern: each step \( \div 2 \) \( 2^1 = 2 \xrightarrow{\div 2} 2^0 = \mathbf{1} \)
