Power Rules Playbook Worksheet Power Rules Playbook
Guided Practice • Product, Power, Quotient, & Zero Laws
Unit: Laws of Exponents
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Part 1 Product of Powers Rule
When multiplying powers with the same base , add the exponents: \( a^m \cdot a^n = a^{m+n} \)
Model: \( x^3 \cdot x^2 = x^{3+2} = \mathbf{x^5} \)
\( x^4 \cdot x^5 \) Expand & add
Step: \( x^{\underline{\hspace{0.6cm}} + \underline{\hspace{0.6cm}}} \)
Ans:
\( 3^2 \cdot 3^4 \) Keep the base
Step: \( 3^{\underline{\hspace{0.6cm}} + \underline{\hspace{0.6cm}}} \)
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\( y^7 \cdot y \) Tip: \( y = y^1 \)
Step: \( y^{\underline{\hspace{0.6cm}} + 1} \)
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\( a^3 \cdot a^5 \cdot a^2 \) Three factors
Step: \( a^{3+5+2} \)
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\( (2m^3) \cdot (4m^6) \) Tip: Multiply coefficients: \( (2 \cdot 4) \), then add powers of \( m \)
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Part 2 Power of a Power Rule
When raising a power to another power, multiply the exponents: \( (a^m)^n = a^{m \cdot n} \)
Model: \( (x^4)^3 = x^{4 \cdot 3} = \mathbf{x^{12}} \)
\( (w^3)^4 \) Multiply powers
Step: \( w^{\underline{\hspace{0.6cm}} \cdot \underline{\hspace{0.6cm}}} \)
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\( (5^2)^5 \) Leave in power form
Step: \( 5^{2 \cdot 5} \)
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\( (k^6)^3 \) Quick check
Step: \( k^{6 \cdot 3} \)
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\( (b^5)^2 \) Multiply exponents
Step: \( b^{5 \cdot 2} \)
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\( (3b^4)^3 \) Rule: Apply exponent to everything inside: \( (3^3) \cdot (b^4)^3 \)
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Remember: Exponent rules only apply when expressions have identical base values. Page 1 of 2
Power Rules Playbook • Quotient & Zero Exponent Laws
Student Name: Page 2 of 2
Part 3 Quotient of Powers Rule
When dividing powers with the same base , subtract the bottom exponent: \( \frac{a^m}{a^n} = a^{m-n} \)
Model: \( \frac{x^7}{x^3} = x^{7-3} = \mathbf{x^4} \)
\( \frac{y^9}{y^4} \) Subtract exponents
Step: \( y^{\underline{\hspace{0.6cm}} - \underline{\hspace{0.6cm}}} \)
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\( \frac{6^{10}}{6^7} \) Keep the base 6
Step: \( 6^{10 - 7} \)
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\( \frac{m^8}{m} \) Tip: denominator is \( m^1 \)
Step: \( m^{8 - 1} \)
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\( \frac{p^8}{p^3} \) Subtract exponents
Step: \( p^{8 - 3} \)
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\( \frac{18c^7}{6c^2} \) Tip: Divide numbers \( (18 \div 6) \), then subtract exponents \( c^{7-2} \)
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Part 4 Zero Exponent Rule
Any non-zero base raised to the power of zero equals 1 : \( a^0 = 1 \quad (a \neq 0) \)
Why? \( \frac{x^3}{x^3} = x^{3-3} = x^0 = \mathbf{1} \)
\( 9^0 \) Numerical base
Rule: \( a^0 = 1 \)
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\( y^0 \) Variable base
Rule: \( y^0 = \underline{\hspace{0.6cm}} \)
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\( (4x)^0 \) Group in parenthesis
Step: Entire base to power 0
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\( 5 \cdot a^0 \) Watch: only \(a\) has power 0
Step: \( 5 \cdot 1 \)
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\( \frac{x^5}{x^5} \) Subtract exponents: \( x^{5-5} = x^0 = \underline{\hspace{0.5cm}} \)
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Exponents Checklist:
Product: Add Power: Multiply Quotient: Subtract Zero Rule: \(a^0 = 1\)
Power Rules Playbook • Product, Power of a Power, Quotient, and Zero Exponent Rules Page 2 of 2
Power Rules Answer Key Power Rules Playbook
Teacher Answer Key
Full Solutions • Product, Power, Quotient, & Zero Laws
Complete Solutions (Q1–20)
Total: 20 Questions (5 pts each = 100 pts) All intermediate steps and final simplified forms provided
Part 1 Product of Powers Rule Answers • \( a^m \cdot a^n = a^{m+n} \)
Add exponents with identical base
\( x^4 \cdot x^5 \) Add: \( 4 + 5 \)
Step: \( x^{4+5} \)
Ans: \( x^9 \)
\( 3^2 \cdot 3^4 \) Add: \( 2 + 4 \)
Step: \( 3^{2+4} \)
Ans: \( 3^6 \) (or 729)
\( y^7 \cdot y \) Misconception: \( y = y^1 \)
Step: \( y^{7+1} \)
Ans: \( y^8 \)
\( a^3 \cdot a^5 \cdot a^2 \) Add all three
Step: \( a^{3+5+2} \)
Ans: \( a^{10} \)
\( (2m^3) \cdot (4m^6) \) Step: Multiply coefficients \( (2 \cdot 4 = 8) \), add powers \( (3 + 6 = 9) \)
Ans: \( 8m^9 \)
Part 2 Power of a Power Rule Answers • \( (a^m)^n = a^{m \cdot n} \)
Multiply the exponents
\( (w^3)^4 \) Multiply: \( 3 \cdot 4 \)
Step: \( w^{3 \cdot 4} \)
Ans: \( w^{12} \)
\( (5^2)^5 \) Multiply: \( 2 \cdot 5 \)
Step: \( 5^{2 \cdot 5} \)
Ans: \( 5^{10} \)
\( (k^6)^3 \) Multiply: \( 6 \cdot 3 \)
Step: \( k^{6 \cdot 3} \)
Ans: \( k^{18} \)
\( (b^5)^2 \) Multiply: \( 5 \cdot 2 \)
Step: \( b^{5 \cdot 2} \)
Ans: \( b^{10} \)
\( (3b^4)^3 \) Step: Apply power to coefficient and variable: \( 3^3 \cdot (b^4)^3 = 27b^{12} \)
Ans: \( 27b^{12} \)
Grading Tip: Look out for Q5 and Q10 where students often mistakenly do \(2+4\) or \(3 \cdot 3\) instead of evaluating powers on coefficients.
Power Rules Playbook • Teacher Solutions Page 1 of 2
Power Rules Playbook • Quotient & Zero Exponent Answers
Page 2 Answers Page 2 of 2
Part 3 Quotient of Powers Rule Answers • \( \frac{a^m}{a^n} = a^{m-n} \)
Subtract bottom exponent
\( \frac{y^9}{y^4} \) Subtract: \( 9 - 4 \)
Step: \( y^{9-4} \)
Ans: \( y^5 \)
Zero Exponent Practice Worksheet Zero Exponent Practice
10-Question Guided Mastery • Properties of the Zero Exponent
10 Questions • Guided Practice
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Core Rule The Zero Exponent Property
Any non-zero quantity raised to the power of 0 equals 1 : \( a^0 = 1 \quad (a \neq 0) \)
Parentheses Matter!
\( (4x)^0 = \mathbf{1} \) ≠ \( 4x^0 = 4(1) = \mathbf{4} \)
\( 15^0 \)
Numerical base
Property: \( a^0 = 1 \)
Ans:
\( m^0 \)
Variable base
Property: \( m^0 = \underline{\hspace{0.5cm}} \)
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\( 6x^0 \)
No parentheses
Step: \( 6 \cdot (x^0) = 6 \cdot 1 \)
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\( (6x)^0 \)
Group in parentheses
Step: \( (\text{entire group})^0 \)
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\( y^5 \cdot y^0 \)
Product Property
Add powers: \( y^{5 + 0} \)
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\( \frac{x^8}{x^8} \)
Quotient Property
Subtract: \( x^{8 - 8} = x^0 \)
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\( (a^4)^0 \)
Power of a Power
Multiply: \( a^{4 \cdot 0} = a^0 \)
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\( 3a^0 b^4 \)
Two variables
Replace \( a^0 \): \( 3 \cdot 1 \cdot b^4 \)
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\( \frac{14k^7}{2k^7} \)
Divide numbers
Step: \( (14 \div 2) \cdot k^{7-7} = 7k^0 \)
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\( (2x^0 y^3)^2 \)
Combo challenge
Step: \( (2 \cdot 1 \cdot y^3)^2 = (2y^3)^2 \)
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Concept Check: True or False?
Circle T or F
A. \( -5^0 \) and \( (-5)^0 \) have the exact same value. [ T / F ]
B. Any expression inside parentheses to the 0 power equals 1. [ T / F ]
Zero Exponent Practice • Complete all 10 problems showing steps Page 1 of 1
Zero Exponent Answer Key Zero Exponent Practice
Teacher Answer Key
10-Question Complete Solutions • Scoring Guide • Misconception Notes
10 Questions • 100 Pts Total
10 Points Per Question (5 pts work shown + 5 pts final answer) Core takeaway: \( a^0 = 1 \quad (a \neq 0) \)
Key Misconception: Watch questions 3 & 4. Students frequently confuse \( c \cdot x^0 \) with \( (cx)^0 \).
\( (cx)^0 = 1 \) vs \( cx^0 = c \)
\( 15^0 \)
Non-zero base
Step: \( a^0 = 1 \)
Ans: \( 1 \)
\( m^0 \)
Variable base
Step: \( m^0 = 1 \)
Ans: \( 1 \)
\( 6x^0 \)
Only \(x\) is to 0
Step: \( 6 \cdot (1) = 6 \)
Ans: \( 6 \)
\( (6x)^0 \)
Parentheses power
Step: \( (6x)^0 = 1 \)
Ans: \( 1 \)
\( y^5 \cdot y^0 \)
Product Property
Step: \( y^{5+0} = y^5 \)
Ans: \( y^5 \)
\( \frac{x^8}{x^8} \)
Quotient Property
Step: \( x^{8-8} = x^0 = 1 \)
Ans: \( 1 \)
\( (a^4)^0 \)
Power of a Power
Step: \( a^{4 \cdot 0} = a^0 = 1 \)
Ans: \( 1 \)
\( 3a^0 b^4 \)
Substitute \( a^0 = 1 \)
Step: \( 3 \cdot 1 \cdot b^4 = 3b^4 \)
Ans: \( 3b^4 \)
\( \frac{14k^7}{2k^7} \)
Coefficients divide
Step: \( 7 \cdot k^{7-7} = 7 \cdot 1 = 7 \)
Ans: \( 7 \)
\( (2x^0 y^3)^2 \)
Two-step combo
Step: \( (2 \cdot 1 \cdot y^3)^2 = 2^2 y^6 = 4y^6 \)
Ans: \( 4y^6 \)
Concept Check Solutions & Explanations:
Answer Key
A. \( -5^0 \) and \( (-5)^0 \) FALSE
\( -5^0 = -(5^0) = -1 \), but \( (-5)^0 = 1 \). Parentheses govern the negative sign!
B. Entire group to 0 equals 1 TRUE
For any non-zero expression, \( (\text{expression})^0 = 1 \) always holds true.
Zero Exponent Practice • Teacher Edition Solutions Complete Page 1 of 1
