Question

simplify.
\\(\frac{4}{2x^2} + \frac{1}{3x}\\)
\\(\frac{x + ?}{\square x^{\square}}\\)

Explanation:

Step 1: Simplify the first fraction

Simplify \(\frac{4}{2x^{2}}\) by dividing 4 by 2.
\(\frac{4}{2x^{2}}=\frac{2}{x^{2}}\)

Step 2: Find a common denominator

To add \(\frac{2}{x^{2}}\) and \(\frac{1}{3x}\), the least common denominator (LCD) of \(x^{2}\) and \(3x\) is \(3x^{2}\).

Step 3: Rewrite fractions with LCD

Rewrite \(\frac{2}{x^{2}}\) with denominator \(3x^{2}\): \(\frac{2}{x^{2}}\times\frac{3}{3}=\frac{6}{3x^{2}}\)
Rewrite \(\frac{1}{3x}\) with denominator \(3x^{2}\): \(\frac{1}{3x}\times\frac{x}{x}=\frac{x}{3x^{2}}\)

Step 4: Add the fractions

Add \(\frac{6}{3x^{2}}\) and \(\frac{x}{3x^{2}}\): \(\frac{6 + x}{3x^{2}}\)
We can also rewrite the denominator as \(3x\times x\) (but the main numerator part for the "x + [?]" is \(x + 6\) and the denominator structure here, but looking at the given format \(\frac{x+[?]}{[]x^{[]}}\), from our result \(\frac{x + 6}{3x^{2}}\), so the "?" is 6, the first box in denominator is 3, and the exponent on x is 2. But focusing on the numerator's "x + [?]", the "?" is 6.

Answer:

6 (and for the denominator boxes, first is 3, exponent on x is 2, but the question's "?" is 6)