Simplifying Algebraic Expressions with Exponents
Step-by-Step Examples

This collection of intermediate algebra problems focuses on simplifying expressions using the rules and properties of exponents. Detailed solutions are provided at the bottom of the page to help reinforce your understanding.

Basic Exponent Rules for Simplifying Expressions

• Product of Powers Rule:
  \[ a^m \cdot a^n = a^{m+n} \]
• Quotient of Powers Rule:
  \[ \dfrac{a^m}{a^n} = a^{m-n} \quad \text{(for } a \ne 0 \text{)} \]
• Power of a Power Rule:
  \[ (a^m)^n = a^{m \cdot n} \]
• Power of a Product Rule:
  \[ (ab)^m = a^m \cdot b^m \]
• Power of a Quotient Rule:
  \[ \left( \dfrac{a}{b} \right)^m = \dfrac{a^m}{b^m} \quad \text{(for } b \ne 0 \text{)} \]
• Zero Exponent Rule:
  \[ a^0 = 1 \quad \text{(for } a \ne 0 \text{)} \]
• Negative Exponent Rule:
  \[ a^{-m} = \dfrac{1}{a^m} \quad \text{(for } a \ne 0 \text{)} \]

1. Simplify the following expressions and write them with positive exponents.
   a) \[x^2 \cdot x^3\]
   b) \[a^{-2} \cdot a^{-3}\]
   c) \[\dfrac{a^5}{a^2}\]
   d) \[\dfrac{a^{-3}}{a^7}\]
   e) \[(x^2)^3\]
   f) \[(x^{-2})^5\]
   g) \[(x^2 y^3)^3\]
   h) \[(x^{-2} y^3)^8\]
2. Simplify the following expressions and write them with positive exponents.
   a) \[\left(\dfrac{y^4}{y^2}\right)^3\]
   b) \[\left(\dfrac{y^4}{y^{-3}}\right)^{-2}\]
   c) \[\left(\dfrac{y^4 x^{12}}{y^2}\right)^3\]
   d) \[\left(\dfrac{y^{12} x^{56}}{y^3}\right)^0\]
   e) \[\dfrac{(x^2 y^4)^2}{(x y^{-2})^3}\]
3. Simplify the following expressions and write them with positive exponents.
   a) \[\dfrac{(x y^2)^2 (x^2 y^2)^2}{(x y)^4}\]
   b) \[\left((a b^2)^3 (a^2 b^2)^4\right)^2\]
   c) \[\dfrac{(x^2 y^3)^0 (x^4 y^3)^4}{(x y^2)^3}\]
   d) \[\dfrac{(x^3 y^{-2})^2 (x^{-2} y^3)^4}{(x^{-2} y^{-2})^3}\]

1. a) \[x^2 \cdot x^3 = x^{2 + 3} = x^5\] b) \[a^{-2} \cdot a^{-3} = a^{-2 - 3} = a^{-5} = \dfrac{1}{a^5}\] c) \[\dfrac{a^5}{a^2} = a^{5 - 2} = a^3\] d) \[\dfrac{a^{-3}}{a^7} = a^{-3 - 7} = a^{-10} = \dfrac{1}{a^{10}}\] e) \[(x^2)^3 = x^{2 \cdot 3} = x^6\] f) \[(x^{-2})^5 = x^{-2 \cdot 5} = x^{-10} = \dfrac{1}{x^{10}}\] g) \[(x^2 y^3)^3 = x^{2 \cdot 3} y^{3 \cdot 3} = x^6 y^9\] h) \[(x^{-2} y^3)^8 = x^{-2 \cdot 8} y^{3 \cdot 8} = x^{-16} y^{24} = \dfrac{y^{24}}{x^{16}}\]
2. a) \[\left( \dfrac{y^4}{y^2} \right)^3 = \dfrac{y^{4 \cdot 3}}{y^{2 \cdot 3}} = \dfrac{y^{12}}{y^6} = y^6\] b) \[\left( \dfrac{y^4}{y^{-3}} \right)^{-2} = \dfrac{y^{4 \cdot (-2)}}{y^{-3 \cdot (-2)}} = \dfrac{y^{-8}}{y^6} = \dfrac{1}{y^{14}}\] c) \[\left( \dfrac{y^4 x^{12}}{y^2} \right)^3 = \dfrac{(y^4 x^{12})^3}{y^{2 \cdot 3}} = \dfrac{y^{12} x^{36}}{y^6} = y^6 x^{36}\] d) \[\left( \dfrac{y^{12} x^{56}}{y^3} \right)^0 = 1\] e) \[\dfrac{(x^2 y^4)^2}{(x y^{-2})^3} = \dfrac{x^{4} y^8}{x^3 y^{-6}} = x y^{14}\]
3. a) \[\dfrac{(x y^2)^2 (x^2 y^2)^2}{(x y)^4} = \dfrac{x^2 y^4 \cdot x^4 y^4}{x^4 y^4} = \dfrac{x^6 y^8}{x^4 y^4} = x^2 y^4\] b) \[\left( (a b^2)^3 (a^2 b^2)^4 \right)^2 = (a b^2)^6 (a^2 b^2)^8 = a^6 b^{12} \cdot a^{16} b^{16} = a^{22} b^{28}\] c) \[\dfrac{(x^2 y^3)^0 (x^4 y^3)^4}{(x y^2)^3} = \dfrac{1 \cdot x^{16} y^{12}}{x^3 y^6} = x^{13} y^6\] d) \[\dfrac{(x^3 y^{-2})^2 (x^{-2} y^3)^4}{(x^{-2} y^{-2})^3} = \dfrac{x^6 y^{-4} \cdot x^{-8} y^{12}}{x^{-6} y^{-6}} = \dfrac{x^{-2} y^8}{x^{-6} y^{-6}} = x^4 y^{14}\]

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