Simplify: $\sqrt{48 x^2 y^3}$
We want to simplify the expression \(\sqrt{48x^2y^3}\). For the square root to be defined as a real number, the term inside (the radicand) must be non-negative. This means \(48x^2y^3 \geq 0\). Since \(x^2 \geq 0\) for any real number \(x\), we must have \(y^3 \geq 0\), which implies \(y \geq 0\). Given this constraint, we can proceed to simplify the radical expression by factoring the components.
First, find the prime factorization of 48: \(48 = 16 \times 3 = 4^2 \times 3\). Now, substitute this into the expression:
\[\sqrt{48x^2y^3} = \sqrt{16 \cdot 3 \cdot x^2 \cdot y^2 \cdot y}.\]
We can use the property \(\sqrt{ab} = \sqrt{a}\sqrt{b}\) to separate the square root into parts:
\[\sqrt{48x^2y^3} = \sqrt{16} \cdot \sqrt{3} \cdot \sqrt{x^2} \cdot \sqrt{y^2} \cdot \sqrt{y}.\]
Now, simplify each term separately:
Since we established in Step 1 that \(y \geq 0\), we have \(|y| = y\).
Combining these, we get:
\[\sqrt{48x^2y^3} = 4 \cdot |x| \cdot y \cdot \sqrt{3y} = 4|x|y\sqrt{3y}.\]
Because \(|x|y = |x||y| = |xy|\) (since \(y \geq 0\)), we can write this as:
\[4|xy|\sqrt{3y}.\]
Simplifying radical expressions is a fundamental skill in algebra with many real-world applications. For example, in physics, when calculating the velocity of an object in a circular motion or determining the distance between two points on a graph, you often encounter square roots. If you are designing a ramp or calculating the hypotenuse of a triangular support beam in architecture, you might end up with an expression like \(\sqrt{48x^2y^3}\). Simplifying such expressions allows you to compare different results more easily, make faster calculations, and interpret the physical meaning of the formulas more clearly.