- Simplify the first term \(\sqrt{5x^2} = x\sqrt{5}\) since \(x \geq 0\).
- Recognize the expression as a binomial square \((A + B)^2 = A^2 + 2AB + B^2\).
- Calculate the expansion: \((4x^2\sqrt{5})^2 + 2(4x^2\sqrt{5})(2x^2\sqrt{6}) + (2x^2\sqrt{6})^2\).
- Simplify to reach the final form \(\boxed{104x^4 + 16x^4\sqrt{30}}\).
Explanation
- Simplifying the expression within the square
We start by simplifying the given expression \(\left(4x\sqrt{5x^2} + 2x^2\sqrt{6}\right)^2\) given that \(x \geq 0\). First, let's look at the first term inside the square, \(4x\sqrt{5x^2}\). Since \(x \geq 0\), we know that \(\sqrt{x^2} = x\). Therefore, we can simplify this term as follows:
\[4x\sqrt{5x^2} = 4x(x\sqrt{5}) = 4x^2\sqrt{5}.\]
Now the original expression is \(\left(4x^2\sqrt{5} + 2x^2\sqrt{6}\right)^2\).
- Expanding the square of the binomial
We apply the algebraic identity for the square of a binomial: \((A + B)^2 = A^2 + 2AB + B^2\). Here, we have \(A = 4x^2\sqrt{5}\) and \(B = 2x^2\sqrt{6}\). We calculate each part separately:
- \(A^2 = (4x^2\sqrt{5})^2 = 16x^4 \cdot 5 = 80x^4\).
- \(B^2 = (2x^2\sqrt{6})^2 = 4x^4 \cdot 6 = 24x^4\).
- \(2AB = 2(4x^2\sqrt{5})(2x^2\sqrt{6}) = 16x^4\sqrt{5 \cdot 6} = 16x^4\sqrt{30}\).
- Final calculation and result identification
Now, we sum the calculated parts to obtain the final expression:
\[A^2 + 2AB + B^2 = 80x^4 + 16x^4\sqrt{30} + 24x^4.\]
Combining the like terms \(80x^4\) and \(24x^4\), we get:
\[(80 + 24)x^4 + 16x^4\sqrt{30} = 104x^4 + 16x^4\sqrt{30}.\]
Comparing this result to the given options, we find it matches the fourth option. Thus, the simplified form of the expression is \(104x^4 + 16x^4\sqrt{30}\).
Examples
Expanding algebraic expressions with radicals is crucial in engineering and physics, where complex formulas often appear. For example, in calculating the kinetic energy or potential energy of particles, one might encounter squares of expressions involving physical constants and variables. Knowing how to correctly expand and simplify these terms helps in reducing complexity and accurately determining physical quantities. This technique is a fundamental step toward mastering more advanced algebraic manipulations used in professional calculations.