Question

What will come in the place of question mark in the following question ?
\[ \sqrt{726} + \sqrt{294} + \sqrt{1176} + \sqrt{486} - \sqrt{600} = (?) \]

• A. \(31\sqrt{2}\)
• B. \(31\sqrt{6}\)
• C. \(34\sqrt{6}\)
• D. \(34\sqrt{2}\)

Hint:

When dealing with multiple large radical terms, look at the options.
The options suggest that the final answer is a multiple of either \(\sqrt{2}\) or \(\sqrt{6}\).
Test the smallest term first, such as \(294\), and divide by \(6\), which gives \(49\) (a perfect square).
This immediately alerts you that \(\sqrt{6}\) is likely the common radical factor for all terms, which dramatically speeds up factorization.

Answer 1

The Correct Option is B

Step 1: A quicker path, estimate first.
Instead of factoring every surd from scratch, we can get a fast check by finding the approximate decimal value of the whole expression and matching it to an option. This works nicely here because every option is written as a whole number times $\sqrt{2}$ or $\sqrt{6}$, so a decimal comparison pins down the answer quickly.
Step 2: Work out the decimal values.
Using $\sqrt{726} \approx 26.94$, $\sqrt{294} \approx 17.15$, $\sqrt{1176} \approx 34.29$, $\sqrt{486} \approx 22.05$ and $\sqrt{600} \approx 24.49$, the expression becomes \[ 26.94 + 17.15 + 34.29 + 22.05 - 24.49 \approx 75.94 \]
Step 3: Match this to the options.
Since $\sqrt{6} \approx 2.449$, we check $31\sqrt{6} \approx 31 \times 2.449 \approx 75.94$, which lines up exactly with our sum. The other options, such as $31\sqrt{2} \approx 43.8$ or $34\sqrt{6} \approx 83.3$, are nowhere close.
Step 4: Confirm with exact values.
Each radicand is 6 times a perfect square: $726 = 6 \times 121$, $294 = 6 \times 49$, $1176 = 6 \times 196$, $486 = 6 \times 81$, $600 = 6 \times 100$, giving $11\sqrt6 + 7\sqrt6 + 14\sqrt6 + 9\sqrt6 - 10\sqrt6 = 31\sqrt6$, exactly matching our estimate.
\[ \boxed{31\sqrt{6}} \]