Simplify the expression: $\frac{\sqrt{50 x^4 y^2 z}}{2 \sqrt{5}}$
To simplify the expression \(\frac{\sqrt{50 x^4 y^2 z}}{2 \sqrt{5}}\), we first examine the structure of the expression. We have a quotient of two square roots, and we can utilize the property of radicals that states \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\) for \(a, b > 0\). This allows us to combine the terms under a single square root. We also assume that \(x, y, z\) are such that the expression is well-defined, specifically \(z \geq 0\).
Using the property \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\), we can rewrite the expression as:
\[\frac{\sqrt{50 x^4 y^2 z}}{2 \sqrt{5}} = \frac{1}{2} \sqrt{\frac{50 x^4 y^2 z}{5}}\]
Next, we simplify the numerical fraction inside the radical:
\[\frac{50}{5} = 10\]
So, the expression simplifies to:
\[\frac{1}{2} \sqrt{10 x^4 y^2 z}\]
Now we can extract the perfect square variables from inside the square root. We know that for any real numbers \(x\) and \(y\), \(\sqrt{x^4} = x^2\) and \(\sqrt{y^2} = |y|\). Applying these properties, we pull these terms out of the square root:
\[\sqrt{10 x^4 y^2 z} = \sqrt{x^4} \cdot \sqrt{y^2} \cdot \sqrt{10 z} = x^2 |y| \sqrt{10 z}\]
Substituting this back into our expression, we get:
\[\frac{1}{2} \cdot x^2 |y| \sqrt{10 z}\]
This can also be written as:
\[\frac{x^2 |y| \sqrt{10 z}}{2}\]
Simplifying radicals is a fundamental skill used in physics and engineering, particularly when dealing with formulas like the distance formula in coordinate geometry or the magnitude of a vector. For example, when calculating the velocity of an object moving in two dimensions, you might end up with an expression like \(\sqrt{v_x^2 + v_y^2}\). Simplifying these expressions is crucial for performing further calculations efficiently and maintaining precision in scientific modeling. Mastering these algebraic manipulations helps streamline complex equations in many technical fields.