Question 2 (Multiple Choice, Worth 1 point): (06.03 MC) Choose the correct simplification of the expression \((5xy^5)^2(y^3)^4\). A. \(25x^2y^{22}\) B. \(10x^2y^{22}\) C. \(25x^3y^{14}\) D. \(10x^3y^{14}\) Question 3 (Multiple Choice, Worth 1 point): (06.05 MC) Aurora is selling tickets to a carnival. The function \(f(x) = 0.5x\) represents the amount of money Aurora earns per ticket, given that \(x\) is the number of tickets she sells. The function \(g(x) = 8x\) represents the number of tickets Aurora sells per hour, where \(x\) is the number of hours she works. Find \(f(g(x))\) and explain what it represents.

For Question 2, we simplify the expression \((5xy^5)^2(y^3)^4\) using the properties of exponents:

Firstly, \((5xy^5)^2\) involves squaring all factors inside the parentheses. This means: - The coefficient 5 is squared to become \(5^2 = 25\), - The \(x\) is squared to become \(x^2\), - And \(y^5\) is squared to become \(y^{5 \cdot 2} = y^{10}\).

So, \((5xy^5)^2 = 25x^2y^{10}\).

Secondly, we need to consider \((y^3)^4\). Raising a power to a power means you multiply the exponents: - \(y^3\) raised to the 4th power becomes \(y^{3 \cdot 4} = y^{12}\).

So, \((y^3)^4 = y^{12}\).

Now, we multiply these together: \(25x^2y^{10} \cdot y^{12}\)

When multiplying powers with the same base, we add the exponents: \(25x^2y^{10+12} = 25x^2y^{22}\)

Therefore, the correct answer is A. \(25x^2y^{22}\)

As for Question 3, we want to find \(f(g(x))\), which is a composition of functions. You find this by applying the function \(g\) first and then applying \(f\) to the result.

\(g(x) = 8x\), and \(f(x) = 0.5x\).

\(f(g(x)) = f(8x) = 0.5(8x) = 4x\)

The function \(f(g(x))\) represents the amount of money Aurora earns from selling tickets per hour. Specifically, she earns 4 dollars per hour (since \(x\) represents the number of hours she works).