What is \(\frac{{{6^2} + \;{7^2} + \;{8^2} + \;{9^2} + \;{{10}^2}}}{{\sqrt {7\; + \;4\sqrt 3 \;} - \;\sqrt {4\; + \;2\sqrt 3 } }}\) equal to?
330
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:
| Option | Value |
|---|---|
| 1 | 330 |
| 2 | 340 |
| 3 | 355 |
| 4 | 366 |
Our calculated value is 330, which matches Option 1.
Revision Table: Key Calculations
| Part | Calculation | Result |
|---|---|---|
| Numerator (Sum of Squares) | \(6^2 + 7^2 + 8^2 + 9^2 + 10^2\) | \(36 + 49 + 64 + 81 + 100 = 330\) |
| Denominator Term 1 | \( \sqrt {7\; + \;4\sqrt 3 } = \sqrt{7 + 2\sqrt{12}} \) | \( \sqrt{4} + \sqrt{3} = 2 + \sqrt{3} \) |
| Denominator Term 2 | \( \sqrt {4\; + \;2\sqrt 3 } \) | \( \sqrt{3} + \sqrt{1} = \sqrt{3} + 1 \) |
| Denominator (Difference) | \( (2 + \sqrt{3}) - (\sqrt{3} + 1) \) | \( 2 + \sqrt{3} - \sqrt{3} - 1 = 1 \) |
| Final Expression Value | \( \frac{330}{1} \) | \( 330 \) |
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 \( \sqrt{4} + \sqrt{3} \).
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Column I
Column II
a.
Equivalent fraction of \(\frac{7}{12}\) is
i.
Proper fraction
b.
Equivalent fraction of \(\frac{9}{15}\) is
ii.
Improper fraction
c.
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iii.
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d.
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iv.
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