- Simplify the radicals: \(\sqrt{5x^4} = \sqrt{5}x^2\) and \(\sqrt{5x^8} = \sqrt{5}x^4\).
- Group the expression: \(4 \cdot 3 \cdot \sqrt{5} \cdot \sqrt{5} \cdot x^2 \cdot x^2 \cdot x^4\).
- Multiply the coefficients: \(12 \cdot 5 = 60\).
- Multiply the variables: \(x^{2+2+4} = x^8\).
- Final answer: \(\boxed{60 x^8}\).
Explanation
- Analyzing the Expression
The expression to simplify is \(4 x^2 \sqrt{5 x^4} \cdot 3 \sqrt{5 x^8}\) with the constraint \(x
eq 0\). We can break down the expression into its constituent parts: the numerical coefficients, the radicals, and the variable terms. Our objective is to simplify each part using the properties of exponents and radicals.
- Simplifying the Radicals
First, let's simplify the radical terms. We use the property \(\sqrt{ab} = \sqrt{a}\sqrt{b}\) and \(\sqrt{x^{2n}} = |x^n|\). Since \(x^4 \geq 0\) and \(x^8 \geq 0\) for all \(x
eq 0\), we have:
\(\sqrt{5x^4} = \sqrt{5} \cdot \sqrt{x^4} = \sqrt{5} \cdot x^2\)
\(\sqrt{5x^8} = \sqrt{5} \cdot \sqrt{x^8} = \sqrt{5} \cdot x^4\)
Thus, the original expression can be rewritten as:
\(4 x^2 \cdot (\sqrt{5} x^2) \cdot 3 \cdot (\sqrt{5} x^4)\)
- Performing the Multiplication
Next, we group the numerical coefficients and the variables. The expression is:
\((4 \cdot 3 \cdot \sqrt{5} \cdot \sqrt{5}) \cdot (x^2 \cdot x^2 \cdot x^4)\)
Performing the multiplications:
\(4 \cdot 3 = 12\)
\(\sqrt{5} \cdot \sqrt{5} = 5\)
\(12 \cdot 5 = 60\)
For the variables, we use the rule \(x^a \cdot x^b \cdot x^c = x^{a+b+c}\):
\(x^2 \cdot x^2 \cdot x^4 = x^{2+2+4} = x^8\)
Combining these results, the simplified expression is \(60 x^8\).
- Conclusion
Comparing our result, \(60 x^8\), with the given options:
- \(12 x^{10} \sqrt{5}\)
- \(60 x^8\)
- \(35 x^{18}\)
- \(7 x^{10} \sqrt{5}\)
We see that the correct choice is \(60 x^8\). The final answer is \(\boxed{60 x^8}\).
Examples
In fields like physics or engineering, simplifying complex radical expressions is often necessary when calculating values such as energy, distance, or time in formulas involving square roots and powers. For instance, if you are determining the kinetic energy of an object where velocity is represented by a radical involving a variable, simplifying the expression first makes it much easier to perform subsequent calculations or compare results with experimental data. Maintaining algebraic accuracy ensures that your final measurements and predictions are reliable and correct.