Verified Answer :

  • The expression consists of two fractions with a common denominator \(x^2\).
  • Using the rule for adding fractions \(\frac{a}{d} + \frac{b}{d} = \frac{a+b}{d}\), we combine the numerators: \(\frac{2+4}{x^2}\).
  • Simplifying the numerator results in \(\frac{6}{x^2}\).
  • The final answer is \(\boxed{\frac{6}{x^2}}\).

Explanation

  1. Analysis of the Expression

We are asked to simplify the expression \(\frac{2}{x^2} + \frac{4}{x^2}\).
This expression consists of two fractions with the same denominator, \(x^2\).
To add fractions that have a common denominator, we simply add their numerators while keeping the denominator unchanged. The rule is:
\[\frac{a}{d} + \frac{b}{d} = \frac{a + b}{d}\]
where \(a = 2\), \(b = 4\), and \(d = x^2\).

  1. Calculation and Comparison

Applying the rule for adding fractions:
\[\frac{2}{x^2} + \frac{4}{x^2} = \frac{2 + 4}{x^2}\]
Calculating the sum of the numerators: \(2 + 4 = 6\).
Thus, the expression simplifies to:
\[\frac{6}{x^2}\]
Comparing this to the given choices:

  • \(\frac{6}{x^2}\) (Matches)
  • \(\frac{8}{x^4}\) (Incorrect)
  • \(\frac{6}{2 x^2}\) (Incorrect)
  • \(\frac{8}{x^2}\) (Incorrect)

The correct choice is \(\frac{6}{x^2}\).

Examples

In real-life scenarios, adding fractions is fundamental when dealing with common units. For example, if you have two parts of a pizza, each representing \(\frac{2}{8}\) and \(\frac{4}{8}\) of the total, you can add them to see you have \(\frac{6}{8}\) or \(\frac{3}{4}\) of the pizza. Similarly, in engineering or physics, combining rates or components that share the same base variable allows for simplified analysis and efficient calculations.

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