(4 * 18 ^ N)/(3 ^ - 1 * 6 ^ (2n + 1) * 2 ^ (- N))

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Jun 07, 2024

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Simplifying the Complex Expression: (4 * 18 ^ n) / (3 ^ -1 * 6 ^ (2n + 1) * 2 ^ (-n))

In this article, we will explore the simplification of a complex algebraic expression, specifically the expression:

(4 * 18 ^ n) / (3 ^ -1 * 6 ^ (2n + 1) * 2 ^ (-n))

Step 1: Simplify the Numerator

The numerator of the expression is 4 * 18 ^ n. We can rewrite 18 as 2 * 3 ^ 2, which gives us:

4 * (2 * 3 ^ 2) ^ n

Using the power rule of exponentiation, we can rewrite this as:

4 * 2 ^ n * 3 ^ (2n)

Step 2: Simplify the Denominator

The denominator of the expression is 3 ^ -1 * 6 ^ (2n + 1) * 2 ^ (-n). We can rewrite 6 as 2 * 3, which gives us:

3 ^ (-1) * (2 * 3) ^ (2n + 1) * 2 ^ (-n)

Using the power rule of exponentiation, we can rewrite this as:

3 ^ (-1) * 2 ^ (2n + 1) * 3 ^ (2n + 1) * 2 ^ (-n)

Step 3: Combine the Simplified Expressions

Now, we can combine the simplified numerator and denominator:

(4 * 2 ^ n * 3 ^ (2n)) / (3 ^ (-1) * 2 ^ (2n + 1) * 3 ^ (2n + 1) * 2 ^ (-n))

Step 4: Simplify the Final Expression

We can simplify the final expression by canceling out common factors:

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

This is the simplified form of the original expression.

Conclusion

In this article, we have successfully simplified the complex algebraic expression (4 * 18 ^ n) / (3 ^ -1 * 6 ^ (2n + 1) * 2 ^ (-n)). By breaking down the expression into smaller parts and applying the power rule of exponentiation, we were able to simplify the expression to its final form.