Exponent Rules Study Guide
Exponent Rules Study Guide
Algebra 1 • Sections 8.1–8.6
Name:
Date: Period:
Directions: Simplify each expression completely using exponent rules. All final answers must be written with positive exponents only. Show all work in the space provided.
1
\( 10x^3 \left(-2x^4 y^3\right) \left(\frac{3}{5} x y^2\right) \)
Answer:
2
\( (-3)^{-7} \cdot (-3)^2 \cdot (-3)^9 \)
Answer:
3
\( -(14x)^0 \)
Answer:
4
\( \frac{6r^{-4} t^5}{24r^3 t^8 u^{-6}} \)
Answer:
5
\( \left(a^5 b^8\right)^{\frac{3}{4}} \)
Answer:
6
\( \frac{(5^3)^4 \cdot (5^2)^6 \cdot (5^4)^3}{(5^9)^4} \)
Answer:
7
\( \left(6ab^3 c\right)^3 \cdot \left(2a^3 c\right)^{-2} \)
Answer:
8
\( \left(\frac{-8x^2 y^4 z^0}{2x^2 y^4 z^6}\right)^{-2} \)
Answer:
Exponent Rules Study Guide
Page 2
Name:
Directions: Continue simplifying each expression. Ensure all final answers contain only positive exponents.
9
\( \left(5x^4 \cdot 2x^{-5}\right)^{-3} \)
Answer:
10
\( \frac{6x^4 \cdot \left(2x y^{-3}\right)^2}{\left(4x^{-2} y^3\right)^2 \cdot x^2} \)
Answer:
11
\( \left(-3a^3 b^{-2}\right)^3 \cdot \left(2a^{-4} b^5\right)^2 \)
Answer:
12
\( \left(\frac{3a^{-3} b^4}{9a^2 b^{-1}}\right)^{-2} \)
Answer:
13
\( \frac{\left(4x^3 y^{-2}\right)^{-2} \cdot 8x^7 y^2}{2x^{-3} y^5} \)
Answer:
14
\( \frac{\left(-2m^3 n^{-2}\right)^4 \cdot \left(3m^{-1} n^5\right)}{8m^4 n^{-3}} \)
Answer:
15
\( \left(\frac{-2a^4 b^{-3} c^0}{4a^{-2} b^2 c^{-5}}\right)^{-3} \)
Answer:
16
\( \frac{\left(3x^2 y^{-3}\right)^3 \cdot \left(2x^{-1} y^4\right)^2}{\left(6x^{-2} y^2\right)^2 \cdot x^3 y} \)
Answer:
Exponent Rules Answer Key
Exponent Rules Study Guide
Answer Key
Algebra 1 • Teacher Solutions • Page 1 (Q1–Q8)
Full Worked Steps
Standard Requirement: All expressions simplified with positive exponents only. 8 Problems • 2 Pts Each
1
\( 10x^3 \left(-2x^4 y^3\right) \left(\frac{3}{5} x y^2\right) \)
• Coeffs: \( 10 \cdot (-2) \cdot \frac{3}{5} = -20 \cdot \frac{3}{5} = -12 \)
• Product rule: \( x^{3+4+1} = x^8 \); \( y^{3+2} = y^5 \)
Answer \( -12x^8 y^5 \)
2
\( (-3)^{-7} \cdot (-3)^2 \cdot (-3)^9 \)
• Product rule: keep base \( (-3) \), add powers
• Exponent sum: \( -7 + 2 + 9 = 4 \)
• Simplify: \( (-3)^4 = 81 \)
Answer \( 81 \quad \text{or} \quad (-3)^4 \)
3
\( -(14x)^0 \)
• Zero power rule: \( (14x)^0 = 1 \) for \( 14x \neq 0 \)
• Negative sign sits outside the base: \( -(1) = -1 \)
Answer \( -1 \)
4
\( \frac{6r^{-4} t^5}{24r^3 t^8 u^{-6}} \)
• Coeffs: \( \frac{6}{24} = \frac{1}{4} \); \( r^{-4-3} = r^{-7} = \frac{1}{r^7} \)
• \( t^{5-8} = t^{-3} = \frac{1}{t^3} \); \( \frac{1}{u^{-6}} = u^6 \)
Answer \( \frac{u^6}{4r^7 t^3} \)
5
\( \left(a^5 b^8\right)^{\frac{3}{4}} \)
• Power of a power: multiply each power by \( \frac{3}{4} \)
• \( a^{5 \cdot \frac{3}{4}} = a^{\frac{15}{4}} \); \( b^{8 \cdot \frac{3}{4}} = b^6 \)
Answer \( a^{\frac{15}{4}} b^6 \)
6
\( \frac{(5^3)^4 \cdot (5^2)^6 \cdot (5^4)^3}{(5^9)^4} \)
• Numerator: \( 5^{12} \cdot 5^{12} \cdot 5^{12} = 5^{36} \)
• Denominator: \( (5^9)^4 = 5^{36} \)
• Quotient: \( \frac{5^{36}}{5^{36}} = 5^0 = 1 \)
Answer \( 1 \)
7
\( \left(6ab^3 c\right)^3 \cdot \left(2a^3 c\right)^{-2} \)
• Expand: \( 216 a^3 b^9 c^3 \cdot \frac{1}{4 a^6 c^2} = \frac{216 a^3 b^9 c^3}{4 a^6 c^2} \)
• Simplify: \( \frac{216}{4}=54 \); \( a^{3-6}=\frac{1}{a^3} \); \( c^{3-2}=c \)
Answer \( \frac{54b^9 c}{a^3} \)
8
\( \left(\frac{-8x^2 y^4 z^0}{2x^2 y^4 z^6}\right)^{-2} \)
• Inside: \( z^0=1 \); cancel \( x^2, y^4 \); \( \frac{-8}{2} = -4 \rightarrow \frac{-4}{z^6} \)
• Power \( -2 \): \( \left(\frac{z^6}{-4}\right)^2 = \frac{z^{12}}{16} \)
Answer \( \frac{z^{12}}{16} \)
Exponent Rules Study Guide
Answer Key
Algebra 1 • Teacher Solutions • Page 2 (Q9–Q16)
Full Worked Steps
Standard Requirement: All expressions simplified with positive exponents only. 8 Problems • 2 Pts Each
9
\( \left(5x^4 \cdot 2x^{-5}\right)^{-3} \)
• Inside: \( (5 \cdot 2) x^{4-5} = 10x^{-1} = \frac{10}{x} \)
• Power \( -3 \): \( \left(\frac{10}{x}\right)^{-3} = \left(\frac{x}{10}\right)^3 = \frac{x^3}{1000} \)
Answer \( \frac{x^3}{1000} \)
10
\( \frac{6x^4 \cdot \left(2x y^{-3}\right)^2}{\left(4x^{-2} y^3\right)^2 \cdot x^2} \)
• Numerator: \( 6x^4 \cdot 4x^2 y^{-6} = 24x^6 y^{-6} \)
• Denominator: \( 16x^{-4} y^6 \cdot x^2 = 16x^{-2} y^6 \)
• Divide: \( \frac{24}{16} x^{6-(-2)} y^{-6-6} = \frac{3}{2} x^8 y^{-12} \)
Answer \( \frac{3x^8}{2y^{12}} \)
11
\( \left(-3a^3 b^{-2}\right)^3 \cdot \left(2a^{-4} b^5\right)^2 \)
• Expand: \( (-27a^9 b^{-6}) \cdot (4a^{-8} b^{10}) \)
• Coeffs: \( -27 \cdot 4 = -108 \)
• Exponents: \( a^{9-8} = a^1 \); \( b^{-6+10} = b^4 \)
Answer \( -108ab^4 \)
12
\( \left(\frac{3a^{-3} b^4}{9a^2 b^{-1}}\right)^{-2} \)
• Inside: \( \frac{1}{3} a^{-3-2} b^{4-(-1)} = \frac{b^5}{3a^5} \)
• Invert & square: \( \left(\frac{3a^5}{b^5}\right)^2 = \frac{9a^{10}}{b^{10}} \)
Answer \( \frac{9a^{10}}{b^{10}} \)
13
\( \frac{\left(4x^3 y^{-2}\right)^{-2} \cdot 8x^7 y^2}{2x^{-3} y^5} \)
• Numerator: \( \frac{1}{16} x^{-6} y^4 \cdot 8x^7 y^2 = \frac{1}{2} x y^6 \)
• Divide by \( 2x^{-3} y^5 \): \( \frac{1}{4} x^{1-(-3)} y^{6-5} = \frac{1}{4} x^4 y \)
Answer \( \frac{x^4 y}{4} \)
14
\( \frac{\left(-2m^3 n^{-2}\right)^4 \cdot \left(3m^{-1} n^5\right)}{8m^4 n^{-3}} \)
• Numerator: \( 16m^{12} n^{-8} \cdot 3m^{-1} n^5 = 48m^{11} n^{-3} \)
• Divide: \( \frac{48}{8} m^{11-4} n^{-3-(-3)} = 6m^7 n^0 = 6m^7 \)
Answer \( 6m^7 \)
15
\( \left(\frac{-2a^4 b^{-3} c^0}{4a^{-2} b^2 c^{-5}}\right)^{-3} \)
• Inside: \( c^0=1 \); \( -\frac{1}{2} a^{4-(-2)} b^{-3-2} c^5 = -\frac{a^6 c^5}{2b^5} \)
• Power \( -3 \): \( \left(-\frac{2b^5}{a^6 c^5}\right)^3 = -\frac{8b^{15}}{a^{18} c^{15}} \)
Answer \( -\frac{8b^{15}}{a^{18} c^{15}} \)
16
\( \frac{\left(3x^2 y^{-3}\right)^3 \cdot \left(2x^{-1} y^4\right)^2}{\left(6x^{-2} y^2\right)^2 \cdot x^3 y} \)
• Top: \( 27x^6 y^{-9} \cdot 4x^{-2} y^8 = 108x^4 y^{-1} \)
• Bottom: \( 36x^{-4} y^4 \cdot x^3 y = 36x^{-1} y^5 \)
• Divide: \( \frac{108}{36} x^{4-(-1)} y^{-1-5} = 3x^5 y^{-6} \)
Answer \( \frac{3x^5}{y^6} \)
