What is the following product? Assume $y \geq 0$.
$\sqrt{y^3} \cdot \sqrt{y^3}$
The given expression is the product of two square roots: \(\sqrt{y^3} \cdot \sqrt{y^3}\), under the constraint that \(y \geq 0\). We are looking for the equivalent expression among the provided options.
Recall the fundamental property of square roots: for any non-negative real number \(A\), the product \(\sqrt{A} \cdot \sqrt{A}\) is equal to \((\sqrt{A})^2\), which simplifies to \(A\).
In our case, let \(A = y^3\). Since the problem states \(y \geq 0\), it follows that \(y^3 \geq 0\) is also a non-negative real number. Thus, we can apply the identity:
\[\sqrt{y^3} \cdot \sqrt{y^3} = (\sqrt{y^3})^2 = y^3.\]
This step effectively simplifies the given product to \(y^3\).
Comparing our result \(y^3\) with the given options:
We see that our result corresponds to the first option. The product of the two square roots is indeed \(y^3\), so the correct answer is \(\boxed{y^3}\).
This type of simplification is frequently used in physical science and engineering when dealing with power-law relationships, such as when calculating energy or area in geometric models. For example, if the area of a square is \(A = s^2\) and we have an expression for a side length as \(\sqrt{y^3}\), squaring that length to find the area involves calculating \(\sqrt{y^3} \cdot \sqrt{y^3} = y^3\). Understanding how to simplify square roots of powers is essential for algebraic manipulation and solving equations efficiently.