Exponent Rules Study Guide
Module 1 • Lessons 1–5 Mid-Module Review
Laws of Exponents Study Guide
Form ID: A
Algebra / Pre-Algebra 8
Name:
Class / Period:
Date:
Part 1: Rules & Law Reference Review
Review before completing practice problems
Rule Name
Algebraic Form
Key Concept / Meaning
Example
Product of Powers
\(a^m \cdot a^n = a^{m+n}\)
Keep base same, add exponents
\(4^3 \cdot 4^5 = 4^8\)
Quotient of Powers
\(\frac{a^m}{a^n} = a^{m-n}\)
Keep base same, subtract bottom exponent
\(\frac{8^7}{8^3} = 8^4\)
Power of a Power
\((a^m)^n = a^{m \cdot n}\)
Multiply inner and outer exponents
\((x^4)^3 = x^{12}\)
Power of a Product
\((ab)^n = a^n b^n\)
Distribute exponent to every factor
\((2y^5)^3 = 2^3 y^{15}\)
Zero Exponent
\(a^0 = 1 \quad (a \neq 0)\)
Any nonzero base to power 0 equals 1
\((-9)^0 = 1\)
Negative Exponent
\(a^{-n} = \frac{1}{a^n} \quad (a \neq 0)\)
Take reciprocal; exponent becomes positive
\(5^{-2} = \frac{1}{5^2}\)
Common Pitfall to Avoid: Do not multiply bases when applying the Product Rule! For example, \(6^5 \cdot 6^9 = 6^{14}\), not \(36^{14}\). The base stays the same while the exponents combine.
Practice Set A: Product & Quotient Rules
Write each expression using exponential notation.
1 \(6^5 \cdot 6^9\)
Product Rule
Work:
1.)
2 \(z^3 \cdot z^8\)
Product Rule
Work:
2.)
3 \(2^c \cdot 2^w\)
Variable Exponents
Work:
3.)
4 \(\frac{5^7}{5^3}\)
Quotient Rule
Work:
4.)
5 \(\frac{135^{13}}{135^5}\)
Quotient Rule
Work:
5.)
6 \(\frac{j^{14}}{j^2}\)
Quotient Rule
Work:
6.)
Module 1 Review • Integer Exponents & Rules Page 1 of 2 Turn page for Problems 7–15 (Powers, Zero/Negative Exponents & Written Response)
Part 2 & Written Analysis
Powers, Zero & Negative Exponents, and Synthesis
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ID: A
Practice Set B: Power of a Power & Power of a Product
Simplify and write solution using exponential notation.
7 \((3w^7)^2\)
Power of Product
Work:
7.)
8 \((3c^4 d^5)^3\)
Multiple Factors
Work:
8.)
9 \((g^6)^4\)
Power of a Power
Work:
9.)
10 \((z^3)^3\)
Power of a Power
Work:
10.)
Practice Set C: Zero & Negative Exponents
Simplify and write using positive exponents or exact values.
11 \((-4)^0\)
Zero Power
Work:
11.)
12 \(12^0\)
Zero Power
Work:
12.)
13 \(14^{-3}\)
Negative Power
Work:
13.)
14 \(9^{-6}\)
Negative Power
Work:
14.)
Part 4: Written Response & Multi-Step Justification
Extended Response
15
Explain the complete process and rules you would use to simplify the expression: \(\frac{13^5 \cdot 13^4}{13^6}\). Be sure to name each specific exponent rule used at each step and determine the final simplified value in exponential notation.
Mathematical Steps
Step 1: Simplify Numerator
Step 2: Apply Quotient Rule
Final Form:
Written Explanation (Sentences) Name rules and explain order of operations
Self-Check: Kept common bases Converted negative exponents to reciprocals Stated rules in #15
Page 2 of 2
Exponent Rules Answer Key
Teacher Answer Key Module 1 • Lessons 1–5
Laws of Exponents Study Guide — Key
Form ID: A (Master Key)
Algebra / Pre-Algebra 8
Target Skills: Laws of Exponents & Synthesis
Total Questions: 15 Problems (1–6 Front, 7–15 Back)
Grading Scale: 17 Total Points
Part 1: Rules & Law Reference Review
Scoring standard: Exponential notation required
Rule Name
Algebraic Form
Key Concept / Meaning
Example
Product of Powers
\(a^m \cdot a^n = a^{m+n}\)
Keep base same, add exponents
\(4^3 \cdot 4^5 = 4^8\)
Quotient of Powers
\(\frac{a^m}{a^n} = a^{m-n}\)
Keep base same, subtract bottom exponent
\(\frac{8^7}{8^3} = 8^4\)
Power of a Power
\((a^m)^n = a^{m \cdot n}\)
Multiply inner and outer exponents
\((x^4)^3 = x^{12}\)
Power of a Product
\((ab)^n = a^n b^n\)
Distribute exponent to every factor
\((2y^5)^3 = 2^3 y^{15}\)
Zero Exponent
\(a^0 = 1 \quad (a \neq 0)\)
Any nonzero base to power 0 equals 1
\((-9)^0 = 1\)
Negative Exponent
\(a^{-n} = \frac{1}{a^n} \quad (a \neq 0)\)
Take reciprocal; exponent becomes positive
\(5^{-2} = \frac{1}{5^2}\)
Grading Checkpoint: Students must keep bases in exponential form (e.g. \(6^{14}\) not evaluated to a large multi-digit number). Check for common errors where students multiply bases: \(6^5 \cdot 6^9 \neq 36^{14}\).
Practice Set A Answers: Product & Quotient Rules (1 pt each)
Simplified exponential notation
1 \(6^5 \cdot 6^9\)
Product Rule
\(6^{5+9}\)
1.) \(6^{14}\)
2 \(z^3 \cdot z^8\)
Product Rule
\(z^{3+8}\)
2.) \(z^{11}\)
3 \(2^c \cdot 2^w\)
Variable Exponents
\(2^{c+w}\)
3.) \(2^{c+w}\)
4 \(\frac{5^7}{5^3}\)
Quotient Rule
\(5^{7-3}\)
4.) \(5^4\)
5 \(\frac{135^{13}}{135^5}\)
Quotient Rule
\(135^{13-5}\)
5.) \(135^8\)
6 \(\frac{j^{14}}{j^2}\)
Quotient Rule
\(j^{14-2}\)
6.) \(j^{12}\)
