Question

multiply, and then simplify if possible. assume that all variables represent positive real numbers.
$\left(\sqrt{6x} - 2\sqrt{7x}\
ight)\left(\sqrt{6x} - 4\sqrt{7x}\
ight)$

$\left(\sqrt{6x} - 2\sqrt{7x}\
ight)\left(\sqrt{6x} - 4\sqrt{7x}\
ight) = \square$
(simplify your answer. type an exact answer, using radicals as needed. do not factor.)

Explanation:

Step1: Apply the FOIL method

We use the FOIL (First, Outer, Inner, Last) method to multiply the two binomials \((\sqrt{6x}-2\sqrt{7x})(\sqrt{6x}-4\sqrt{7x})\).

• First terms: \(\sqrt{6x} \times \sqrt{6x} = (\sqrt{6x})^2 = 6x\)
• Outer terms: \(\sqrt{6x} \times (-4\sqrt{7x}) = -4\sqrt{6x \times 7x} = -4\sqrt{42x^2} = -4x\sqrt{42}\) (since \(x\) is positive, \(\sqrt{x^2}=x\))
• Inner terms: \(-2\sqrt{7x} \times \sqrt{6x} = -2\sqrt{7x \times 6x} = -2\sqrt{42x^2} = -2x\sqrt{42}\)
• Last terms: \(-2\sqrt{7x} \times (-4\sqrt{7x}) = 8(\sqrt{7x})^2 = 8 \times 7x = 56x\)

Step2: Combine like terms

Now we combine the terms we got from the FOIL method:

\[

$$\begin{align*} &6x - 4x\sqrt{42} - 2x\sqrt{42} + 56x\\ =&(6x + 56x) + (-4x\sqrt{42} - 2x\sqrt{42})\\ =&62x - 6x\sqrt{42} \end{align*}$$

\]

Answer:

\(62x - 6x\sqrt{42}\)