Verified Answer :

  • Factor the radicands: \(28 = 4 \times 7\) and \(63 = 9 \times 7\).
  • Simplify the individual radicals: \(\sqrt{28x} = 2\sqrt{7x}\) and \(\sqrt{63x} = 3\sqrt{7x}\).
  • Combine the simplified radicals to get \(2\sqrt{7x} + 3\sqrt{7x} = 5\sqrt{7x}\).
  • The final simplified expression is \(\boxed{5\sqrt{7x}}\).

Explanation

  1. Understanding the Problem

We are given the expression \(\sqrt{28x} + \sqrt{63x}\), where \(x\) is a positive real number. To simplify this expression, we need to look for common factors within the square roots that are perfect squares. By pulling these factors out, we can simplify each radical term individually. Once the radicals are in a common form, we can combine the terms if they share the same radicand.

  1. Simplifying Each Radical Term

First, we examine the coefficient of \(x\) in the first term, which is 28. We want to find the largest perfect square factor of 28. Checking its factors, we see that \(28 = 4 \times 7\), where 4 is a perfect square. Thus, we can rewrite the first radical:
\[\sqrt{28x} = \sqrt{4 \times 7x} = \sqrt{4} \times \sqrt{7x} = 2\sqrt{7x}\]
Next, we examine the coefficient of \(x\) in the second term, which is 63. We look for the largest perfect square factor of 63. Checking its factors, we see that \(63 = 9 \times 7\), where 9 is a perfect square. Thus, we can rewrite the second radical:
\[\sqrt{63x} = \sqrt{9 \times 7x} = \sqrt{9} \times \sqrt{7x} = 3\sqrt{7x}\]

  1. Combining Like Terms

Now that both terms are expressed in terms of \(\sqrt{7x}\), we can substitute them back into the original expression:
\[\sqrt{28x} + \sqrt{63x} = 2\sqrt{7x} + 3\sqrt{7x}\]
Since these are like terms, we can add their coefficients together:
\[2\sqrt{7x} + 3\sqrt{7x} = (2 + 3)\sqrt{7x} = 5\sqrt{7x}\]
Thus, the simplified expression is \(5\sqrt{7x}\).

Examples

Square root simplification is essential in fields like engineering and physics when calculating lengths or magnitudes from radical expressions. For example, if you are calculating the total distance covered in a two-part journey where the distances are represented by \(\sqrt{28x}\) and \(\sqrt{63x}\) meters, simplifying the expression to \(5\sqrt{7x}\) meters makes it much easier to perform further calculations or estimate the value for different values of \(x\). This process helps in quickly reducing complex formulas to their simplest, most manageable form.