Simplifying Algebraic Expressions with Exponents: Step-by-Step Examples

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:
\[ a^{-m} = \dfrac{1}{a^m} \quad \text{(for } a \ne 0 \text{)} \]

Note: All letters (variables) used in the expressions below represent numbers that are not equal to 0.

Practice Problems & Step-by-Step Solutions

Question 1: Basic Exponent Properties

Problem: Simplify the following expressions and write them with positive exponents:

\( x^2 \cdot x^3 \)
\( a^{-2} \cdot a^{-3} \)
\( \dfrac{a^5}{a^2} \)
\( \dfrac{a^{-3}}{a^7} \)
\( (x^2)^3 \)
\( (x^{-2})^5 \)
\( (x^2 y^3)^3 \)
\( (x^{-2} y^3)^8 \)

Solutions:

\( x^2 \cdot x^3 = x^{2 + 3} = x^5 \)
\( a^{-2} \cdot a^{-3} = a^{-2 - 3} = a^{-5} = \dfrac{1}{a^5} \)
\( \dfrac{a^5}{a^2} = a^{5 - 2} = a^3 \)
\( \dfrac{a^{-3}}{a^7} = a^{-3 - 7} = a^{-10} = \dfrac{1}{a^{10}} \)
\( (x^2)^3 = x^{2 \cdot 3} = x^6 \)
\( (x^{-2})^5 = x^{-2 \cdot 5} = x^{-10} = \dfrac{1}{x^{10}} \)
\( (x^2 y^3)^3 = x^{2 \cdot 3} y^{3 \cdot 3} = x^6 y^9 \)
\( (x^{-2} y^3)^8 = x^{-2 \cdot 8} y^{3 \cdot 8} = x^{-16} y^{24} = \dfrac{y^{24}}{x^{16}} \)

Question 2: Quotients and Powers of Fractions

Problem: Simplify the following expressions and write them with positive exponents:

\( \left(\dfrac{y^4}{y^2}\right)^3 \)
\( \left(\dfrac{y^4}{y^{-3}}\right)^{-2} \)
\( \left(\dfrac{y^4 x^{12}}{y^2}\right)^3 \)
\( \left(\dfrac{y^{12} x^{56}}{y^3}\right)^0 \)
\( \dfrac{(x^2 y^4)^2}{(x y^{-2})^3} \)

Solutions:

\( \left( \dfrac{y^4}{y^2} \right)^3 = \dfrac{y^{4 \cdot 3}}{y^{2 \cdot 3}} = \dfrac{y^{12}}{y^6} = y^6 \)
\( \left( \dfrac{y^4}{y^{-3}} \right)^{-2} = \dfrac{y^{4 \cdot (-2)}}{y^{-3 \cdot (-2)}} = \dfrac{y^{-8}}{y^6} = y^{-14} = \dfrac{1}{y^{14}} \)
\( \left( \dfrac{y^4 x^{12}}{y^2} \right)^3 = \dfrac{(y^4 x^{12})^3}{y^{2 \cdot 3}} = \dfrac{y^{12} x^{36}}{y^6} = y^6 x^{36} \)
\( \left( \dfrac{y^{12} x^{56}}{y^3} \right)^0 = 1 \)
\( \dfrac{(x^2 y^4)^2}{(x y^{-2})^3} = \dfrac{x^{4} y^8}{x^3 y^{-6}} = x^{4-3} y^{8 - (-6)} = x y^{14} \)

Question 3: Advanced Combined Exponent Expressions

Problem: Simplify the following expressions and write them with positive exponents:

Solutions:

\( \dfrac{(x y^2)^2 (x^2 y^2)^2}{(x y)^4} = \dfrac{x^2 y^4 \cdot x^4 y^4}{x^4 y^4} = \dfrac{x^6 y^8}{x^4 y^4} = x^2 y^4 \)
\( \left( (a b^2)^3 (a^2 b^2)^4 \right)^2 = (a^3 b^6 \cdot a^8 b^8)^2 = (a^{11} b^{14})^2 = a^{22} b^{28} \)
\( \dfrac{(x^2 y^3)^0 (x^4 y^3)^4}{(x y^2)^3} = \dfrac{1 \cdot x^{16} y^{12}}{x^3 y^6} = x^{13} y^6 \)
\( \dfrac{(x^3 y^{-2})^2 (x^{-2} y^3)^4}{(x^{-2} y^{-2})^3} = \dfrac{x^6 y^{-4} \cdot x^{-8} y^{12}}{x^{-6} y^{-6}} = \dfrac{x^{-2} y^8}{x^{-6} y^{-6}} = x^4 y^{14} \)

Simplifying Algebraic Expressions with Exponents
Step-by-Step Examples