(a^(2 N+3)*a^((2n+1)(n+2)))/((an^(3))^(2n+1)*a^(n(2n+1)))

Jasonbradley

2 min read

·

Jun 16, 2024

57

Share

Simplifying Exponential Expressions

This article will guide you through simplifying the following complex exponential expression:

(a^(2n+3) * a^((2n+1)(n+2))) / ((an^(3))^(2n+1) * a^(n(2n+1)))

Let's break down the simplification step-by-step:

Utilizing Exponent Rules

1. Product of Powers:

When multiplying exponents with the same base, add the powers:

• a^(2n+3) * a^((2n+1)(n+2)) = a^(2n+3 + (2n+1)(n+2))

2. Power of a Power:

When raising a power to another power, multiply the exponents:

• (an^(3))^(2n+1) = a^(n(3)(2n+1)) = a^(3n(2n+1))

3. Simplifying Exponents:

Expand and simplify the exponents in both numerator and denominator:

• Numerator: a^(2n+3 + (2n+1)(n+2)) = a^(2n+3 + 2n^2 + 5n + 2) = a^(2n^2 + 7n + 5)
• Denominator: a^(3n(2n+1)) * a^(n(2n+1)) = a^(6n^2 + 3n) * a^(2n^2 + n) = a^(8n^2 + 4n)

4. Division of Powers:

When dividing exponents with the same base, subtract the powers:

• (a^(2n^2 + 7n + 5)) / (a^(8n^2 + 4n)) = a^(2n^2 + 7n + 5 - (8n^2 + 4n))

5. Final Simplification:

Simplify the exponent in the final expression:

• a^(2n^2 + 7n + 5 - (8n^2 + 4n)) = a^(-6n^2 + 3n + 5)

Result

Therefore, the simplified form of the given expression is a^(-6n^2 + 3n + 5).