*a%5E((2n%2B1)(n%2B2)))%2F((a%5E(3))%5E(2n%2B1)*a%5E(n(2n%2B1))).jpeg "(a^(2n+3)*a^((2n+1)(n+2)))/((a^(3))^(2n+1)*a^(n(2n+1)))")
Simplifying Exponential Expressions
This article aims to simplify the following exponential expression:
**(a^(2n+3)a^((2n+1)(n+2)))/((a^(3))^(2n+1)a^(n(2n+1)))
Let's break down the process step-by-step:
Applying the Properties of Exponents
To simplify this expression, we will utilize the following properties of exponents:
- Product of powers: a^m * a^n = a^(m+n)
- Power of a power: (a^m)^n = a^(m*n)
Simplifying the Numerator
- *a^(2n+3)a^((2n+1)(n+2)) can be simplified by applying the product of powers rule.
- Expanding the exponent in the second term: (2n+1)(n+2) = 2n^2 + 5n + 2
- Combining the exponents: a^(2n+3) * a^(2n^2 + 5n + 2) = a^(2n^2 + 7n + 5)
Simplifying the Denominator
- *(a^(3))^(2n+1)a^(n(2n+1)) can be simplified by applying the power of a power rule and then the product of powers rule.
- Applying power of a power: (a^(3))^(2n+1) = a^(3*(2n+1)) = a^(6n+3)
- Expanding the exponent in the second term: n(2n+1) = 2n^2 + n
- Combining the exponents: a^(6n+3) * a^(2n^2 + n) = a^(2n^2 + 7n + 3)
Combining the Numerator and Denominator
Now we have: a^(2n^2 + 7n + 5) / a^(2n^2 + 7n + 3)
- Applying the division of powers rule (a^m / a^n = a^(m-n)), we get: a^(2n^2 + 7n + 5 - (2n^2 + 7n + 3))
- Simplifying the exponent: a^(2n^2 + 7n + 5 - 2n^2 - 7n - 3) = a^2
Final Result
Therefore, the simplified expression is a^2.