Understanding the Mathematical Expression

The problem asks us to evaluate a fraction where the numerator is the sum of squares of numbers from 6 to 10, and the denominator involves the difference of two square roots.

The expression is given as: \( \frac{{{6^2} + \;{7^2} + \;{8^2} + \;{9^2} + \;{{10}^2}}}{{\sqrt {7\; + \;4\sqrt 3 \;} - \;\sqrt {4\; + \;2\sqrt 3 } }} \)

Calculating the Numerator: Sum of Squares

The numerator is the sum of the squares of the integers from 6 to 10. We need to calculate each square and then add them up.

• \(6^2 = 6 \times 6 = 36\)
• \(7^2 = 7 \times 7 = 49\)
• \(8^2 = 8 \times 8 = 64\)
• \(9^2 = 9 \times 9 = 81\)
• \(10^2 = 10 \times 10 = 100\)

Now, we sum these values:

\( \text{Numerator} = 36 + 49 + 64 + 81 + 100 \)

\( \text{Numerator} = 85 + 64 + 81 + 100 \)

\( \text{Numerator} = 149 + 81 + 100 \)

\( \text{Numerator} = 230 + 100 \)

\( \text{Numerator} = 330 \)

So, the numerator evaluates to 330.

Simplifying the Denominator: Difference of Square Roots

The denominator is \( \sqrt {7\; + \;4\sqrt 3 \;} - \;\sqrt {4\; + \;2\sqrt 3 } \). We need to simplify each square root term separately.

Simplifying the First Term: \(\sqrt {7\; + \;4\sqrt 3 }\)

We look for a pattern like \( \sqrt{(a+b) + 2\sqrt{ab}} = \sqrt{a} + \sqrt{b} \). The term is \( \sqrt {7\; + \;4\sqrt 3 } \). We can rewrite \( 4\sqrt{3} \) as \( 2 \times 2\sqrt{3} = 2 \times \sqrt{4 \times 3} = 2\sqrt{12} \). So, the expression becomes \( \sqrt{7 + 2\sqrt{12}} \). Now we need two numbers that add up to 7 and multiply to 12. These numbers are 4 and 3. Thus, \( \sqrt{7 + 2\sqrt{12}} = \sqrt{4} + \sqrt{3} = 2 + \sqrt{3} \).

Simplifying the Second Term: \(\sqrt {4\; + \;2\sqrt 3 }\)

This term is already in the form \( \sqrt{(a+b) + 2\sqrt{ab}} \). We need two numbers that add up to 4 and multiply to 3. These numbers are 3 and 1. Thus, \( \sqrt{4 + 2\sqrt{3}} = \sqrt{3} + \sqrt{1} = \sqrt{3} + 1 \).

Calculating the Difference

Now we subtract the second simplified term from the first simplified term:

\( \text{Denominator} = (2 + \sqrt{3}) - (\sqrt{3} + 1) \)

\( \text{Denominator} = 2 + \sqrt{3} - \sqrt{3} - 1 \)

\( \text{Denominator} = (2 - 1) + (\sqrt{3} - \sqrt{3}) \)

\( \text{Denominator} = 1 + 0 \)

\( \text{Denominator} = 1 \)

So, the denominator simplifies to 1.

Evaluating the Full Expression

Now we divide the numerator by the denominator:

\( \frac{{\text{Numerator}}}{{\text{Denominator}}} = \frac{{330}}{{1}} = 330 \)

The value of the given expression is 330.

Comparing with Options

Let's check the given options:

Our calculated value is 330, which matches Option 1.

Revision Table: Key Calculations

Additional Information: Simplifying Square Roots

The technique used to simplify square roots like \( \sqrt{a \pm \sqrt{b}} \) or \( \sqrt{a \pm 2\sqrt{b}} \) is very useful. The general form is \( \sqrt{(x+y) \pm 2\sqrt{xy}} = \sqrt{x} \pm \sqrt{y} \), assuming \( x > y \). To use this, you need to manipulate the term inside the square root to have a ' \(2\) ' multiplying the inner square root. For example, in \( \sqrt{7 + 4\sqrt{3}} \), we changed \( 4\sqrt{3} \) to \( 2 \times 2\sqrt{3} = 2\sqrt{12} \). Then we looked for two numbers that sum to the number outside the inner root (7) and multiply to the number inside the inner root (12). These were 4 and 3, leading to \( \sqrt{4} + \sqrt{3} \).