Simplifying Algebraic Expressions with Exponents

Equations, Inequalities, and Modeling

Simplifying Algebraic Expressions with Exponents

In College Algebra, simplifying expressions involving exponents is a foundational skill. This process uses the laws of exponents to rewrite expressions in their simplest form.

• Exponent Rules: When raising a power to another power, multiply the exponents: $(a^m)^n = a^{mn}$

• Distributive Property of Exponents: For products inside parentheses, apply the exponent to each factor: $(ab)^n = a^n b^n$

Example Problem

Simplify the expression:

$(2b^4)^3 + (-3b^3)^4$

• Step 1: Apply the exponent to each factor inside the parentheses.

• Step 2: Use the exponent rules to simplify.

Solution:

• $(2b^4)^3 = 2^3 (b^4)^3 = 8b^{12}$

• $(-3b^3)^4 = (-3)^4 (b^3)^4 = 81b^{12}$

Final Simplified Expression:

$8b^{12} + 81b^{12} = 89b^{12}$

• Key Point: Combine like terms by adding their coefficients when the variable part is identical.

Additional info: This type of problem is common in early algebra courses and helps reinforce the properties of exponents and combining like terms.