Certainly! Below are some common exponent rules that you might need for your Unit 3 Lesson 12 on exponent rules, along with brief explanations. After that, I’ll provide you with some practice problems to help you prepare for your test.
Exponent Rules
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Product of Powers Rule: \[ a^m \cdot a^n = a^{m+n} \]
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Quotient of Powers Rule: \[ \frac{a^m}{a^n} = a^{m-n} \quad (a \neq 0) \]
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Power of a Power Rule: \[ (a^m)^n = a^{m \cdot n} \]
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Power of a Product Rule: \[ (ab)^n = a^n \cdot b^n \]
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Power of a Quotient Rule: \[ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \quad (b \neq 0) \]
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Zero Exponent Rule: \[ a^0 = 1 \quad (a \neq 0) \]
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Negative Exponent Rule: \[ a^{-n} = \frac{1}{a^n} \quad (a \neq 0) \]
Practice Problems
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Simplify the expression: \[ x^5 \cdot x^3 \]
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Simplify the expression: \[ \frac{y^6}{y^2} \]
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Simplify the expression: \[ (2^3)^2 \]
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Simplify the expression: \[ (3xy)^2 \]
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Evaluate: \[ 4^0 + 2^{-3} \]
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Simplify the expression: \[ \frac{a^{-2}}{a^3} \]
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Simplify the expression: \[ \left(\frac{5}{2}\right)^{-2} \]
Answers to Practice Problems
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\( x^{5+3} = x^8 \)
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\( y^{6-2} = y^4 \)
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\( 2^{3\cdot2} = 2^6 = 64 \)
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\( (3xy)^2 = 3^2 \cdot x^2 \cdot y^2 = 9x^2y^2 \)
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\( 4^0 + 2^{-3} = 1 + \frac{1}{2^3} = 1 + \frac{1}{8} = \frac{8}{8} + \frac{1}{8} = \frac{9}{8} \)
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\( a^{-2-3} = a^{-5} = \frac{1}{a^5} \)
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\( \left(\frac{5}{2}\right)^{-2} = \frac{1}{(\frac{5}{2})^2} = \frac{1}{\frac{25}{4}} = \frac{4}{25} \)
Additional Tips
- Always remember to simplify your final answers.
- When working with negative exponents, think about rewriting them as reciprocals.
- Practice, practice, practice! Make sure to work through plenty of examples to solidify your understanding of these rules.
Good luck on your unit test! If you have any specific questions or need further assistance, feel free to ask!
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