(x^(2n+3)x^((2n+1)(n+2)))/((x^(3))^(2n+1)x^(n(2n+1)))

3 min read Jun 17, 2024

Simplifying the Expression: (x^(2n+3)x^((2n+1)(n+2)))/((x^(3))^(2n+1)x^(n(2n+1)))

This article will guide you through the process of simplifying the given expression:

**(x^(2n+3)x^((2n+1)(n+2)))/((x^(3))^(2n+1)x^(n(2n+1)))

To simplify this expression, we will use the following exponent rules:

Step 1: Simplify the numerator.

Using the product of powers rule, we can combine the terms in the numerator: x^(2n+3) * x^((2n+1)(n+2)) = x^(2n+3 + (2n+1)(n+2))

Step 2: Expand the numerator exponent.

Expanding the expression in the numerator exponent, we get: x^(2n+3 + (2n+1)(n+2)) = x^(2n+3 + 2n^2 + 5n + 2) = x^(2n^2 + 7n + 5)

Step 3: Simplify the denominator.

Applying the power of a power rule to the first term in the denominator: (x^(3))^(2n+1) = x^(3(2n+1)) = x^(6n+3)

Now, using the product of powers rule for the entire denominator: x^(6n+3) * x^(n(2n+1)) = x^(6n+3 + n(2n+1))

Step 4: Expand the denominator exponent.

Expanding the expression in the denominator exponent: x^(6n+3 + n(2n+1)) = x^(6n+3 + 2n^2 + n) = x^(2n^2 + 7n + 3)

Step 5: Combine numerator and denominator.

Now, our expression looks like this: (x^(2n^2 + 7n + 5))/(x^(2n^2 + 7n + 3))

Step 6: Apply the quotient of powers rule.

The quotient of powers rule states: x^m / x^n = x^(m-n)

Applying this to our expression: x^(2n^2 + 7n + 5) / x^(2n^2 + 7n + 3) = x^((2n^2 + 7n + 5) - (2n^2 + 7n + 3))

Step 7: Simplify the final exponent.

Simplifying the exponent in the final step: x^((2n^2 + 7n + 5) - (2n^2 + 7n + 3)) = x^(2)

Therefore, the simplified form of the given expression is: x^2