- Divide the constant coefficients: \(\frac{18}{6} = 3\).
- Simplify the radical fraction: \(\sqrt{\frac{14 x^8}{7 x^4}} = \sqrt{2 x^4}\).
- Simplify the variable radical part: \(\sqrt{2 x^4} = x^2 \sqrt{2}\).
- Multiply the simplified parts to get the final expression: \(3 x^2 \cdot x^2 \sqrt{2} = \boxed{3 x^4 \sqrt{2}}\).
Explanation
- Analysis of the expression
The problem asks us to simplify the algebraic expression \(\frac{18 x^2 \sqrt{14 x^8}}{6 \sqrt{7 x^4}}\) given that \(x
eq 0\). We can break this down by simplifying the constant parts and the radical parts separately.
- Simplifying the radicals
First, simplify the division of the constant coefficients: \(\frac{18}{6} = 3\). Now, the expression becomes \(3 x^2 \frac{\sqrt{14 x^8}}{\sqrt{7 x^4}}\). Using the property of radicals \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\), we can combine the square roots into one: \(\sqrt{\frac{14 x^8}{7 x^4}}\). Simplifying the fraction inside the square root, we get \(\sqrt{2 x^{8-4}} = \sqrt{2 x^4}\).
- Final simplification
Now we simplify the term \(\sqrt{2 x^4}\). Using the product property of square roots, \(\sqrt{2 x^4} = \sqrt{2} \cdot \sqrt{x^4}\). Since \(\sqrt{x^4} = x^2\) (as \(x^2\) is always non-negative), this becomes \(x^2 \sqrt{2}\). Multiplying this by our previous result of \(3 x^2\), we get \(3 x^2 \cdot x^2 \sqrt{2} = 3 x^4 \sqrt{2}\).
- Comparison with options
Comparing our result \(3 x^4 \sqrt{2}\) with the given options:
- \(12 x^4 \sqrt{2}\)
- \(3 x^4 \sqrt{2}\)
- \(3 x^4 \sqrt{7}\)
- \(3 x \sqrt{2}\)
The equivalent expression is \(3 x^4 \sqrt{2}\).
Examples
Understanding how to simplify radical expressions involving variables is fundamental in fields like physics and engineering, where complex formulas often need to be reduced to their simplest form to make calculations manageable. For example, when calculating the velocity of an object or the strength of materials, you might encounter radical terms that, once simplified using these rules, reveal clear relationships between variables, making it easier to predict outcomes or design more efficient systems.