If $y = 5$, then what is the value of $10y\sqrt{y^3 - y^2}$ ?

### Concept & Strategy Perform straight algebraic evaluation by replacing all instances of the variable $y$ with its given value, $5$. Factor or resolve internal values under the radical before final multiplication. Core step: $$ \text{Substitute } y = 5 \text{ into the radical function.} $$ ### Step-by-Step Solution - **Step 1: Substitute $y = 5$ into the statement.** - Expression: $10(5) \times \sqrt{5^3 - 5^2}$ - **Step 2: Simplify components outside and inside the radical.** - Outside product: $10 \times 5 = 50$ - Inside powers: $5^3 = 125$ and $5^2 = 25$ - Subtraction under radical: $125 - 25 = 100$ - **Step 3: Resolve the remaining term.** - Expression is now: $50 \times \sqrt{100}$ - Since $\sqrt{100} = 10$, we evaluate: $50 \times 10 = 500$ ### Exam Strategy & Shortcut Factor out $y^2$ inside the radical first to work with smaller values: $\sqrt{y^3 - y^2} = \sqrt{y^2(y - 1)} = y\sqrt{y - 1}$. Substitute $y = 5$ into this simplified form: $5\sqrt{5 - 1} = 5\sqrt{4} = 5 \times 2 = 10$. Multiply by the initial front term $10y = 50$, yielding $50 \times 10 = 500$. ### Common Pitfall Mistaking $5^3$ as $15$ or $5^2$ as $10$ due to rushing. Always write down exponential expansions carefully if you are prone to simple mental arithmetic slips. ### Final Answer Therefore, the correct answer is 500.