- Identify the expression to simplify: \(\sqrt{128 x^8 y^3 z^9}\).
- Factor the coefficient and powers: \(128 = 8^2 \cdot 2\), \(x^8 = (x^4)^2\), \(y^3 = y^2 \cdot y\), \(z^9 = (z^4)^2 \cdot z\).
- Pull out perfect squares from the radical, using the given constraints \(y \geq 0, z \geq 0\).
- Simplify to reach the final form: \(\boxed{8 x^4 y z^4 \sqrt{2 y z}}\).
Explanation
- Understanding the Problem
We need to simplify the expression \(\sqrt{128 x^8 y^3 z^9}\) given the constraints \(y \geq 0\) and \(z \geq 0\). Our goal is to identify the equivalent expression from the provided choices by extracting perfect square factors from the radical.
- Factoring the Expression
To simplify the radical \(\sqrt{128 x^8 y^3 z^9}\), we begin by factoring the coefficient and the powers of the variables into products of perfect squares and remaining factors.
First, for the numerical coefficient:
\[128 = 64 \times 2 = 8^2 \times 2\]
Next, for the variable powers:
\[x^8 = (x^4)^2\]
\[y^3 = y^2 \times y\]
\[z^9 = z^8 \times z = (z^4)^2 \times z\]
Combining these, we have:
\[\sqrt{128 x^8 y^3 z^9} = \sqrt{8^2 \cdot (x^4)^2 \cdot y^2 \cdot (z^4)^2 \cdot 2 \cdot y \cdot z}\]
- Simplifying the Radical
Using the property \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\) and \(\sqrt{k^2} = |k|\), we can pull the perfect squares out of the radical:
\[\sqrt{8^2 \cdot (x^4)^2 \cdot y^2 \cdot (z^4)^2 \cdot 2 \cdot y \cdot z} = 8 \cdot |x^4| \cdot |y| \cdot |z^4| \cdot \sqrt{2 y z}\]
Given the conditions \(y \geq 0\) and \(z \geq 0\), and knowing that \(x^4 \geq 0\) and \(z^4 \geq 0\) for any real \(x\) and \(z\), we can remove the absolute value signs:
\[8 \cdot x^4 \cdot y \cdot z^4 \cdot \sqrt{2 y z} = 8 x^4 y z^4 \sqrt{2 y z}\]
- Comparing with Options
Comparing our simplified expression \(8 x^4 y z^4 \sqrt{2 y z}\) with the given options, it clearly matches the third option:
\[8 x^4 y z^4 \sqrt{2 y z}\]
Thus, the correct equivalent expression is \(\boxed{8 x^4 y z^4 \sqrt{2 y z}}\).
Examples
Simplifying radicals is a fundamental skill in algebra with many practical applications. For example, in physics, when calculating the magnitude of a vector or the radius of a circle from the area formula, you often end up with complex expressions under a radical. By simplifying these expressions, like converting \(\sqrt{128 x^8 y^3 z^9}\) to \(8 x^4 y z^4 \sqrt{2 y z}\), you make it significantly easier to perform further calculations, compare values, or plug in specific variables for engineering or scientific problems. This systematic approach saves time and reduces the likelihood of errors in complex algebraic manipulations.