Question

Use the laws of exponents to simplify: 27^n * (3)^5 - (3)^(3n-1) * (81) / (9^(2n) * (3^3).

Answer 1

To simplify the expression, use the laws of exponents and combine like terms in the numerator and denominator.

Step-by-step explanation:

To simplify the expression 27^n * (3)^5 - (3)^(3n-1) * (81) / (9^(2n) * (3^3)), we can use the laws of exponents.

1. Start by simplifying the numerator: 27^n * (3)^5 - (3)^(3n-1) * (81)
2. Use the laws of exponents to simplify each term: (27^n)*(3^5) = 3^(3n)*3^2^2*3^3^3^3, (3)^(3n-1)*(81) = 3^(3n)*3^3*3^3
3. Combine like terms in the numerator: 3^(3n)*3^2^2*3^3^3^3 - 3^(3n)*3^3*3^3
4. Simplify the denominator: 9^(2n)*(3^3) = (3^2)^2^2n*(3^3)
5. Combine like terms in the denominator: (3^2)^2^2n*(3^3) = 3^(4n)*3^3
6. Divide the numerator by the denominator: (3^(3n)*3^2^2*3^3^3^3 - 3^(3n)*3^3*3^3) / (3^(4n)*3^3)
7. Use the law of division of exponentials to simplify: (3^(3n)*3^2^2*3^3^3^3 - 3^(3n)*3^3*3^3) / (3^(4n)*3^3) = 3^(3n-22) - 3^(3n-9)