What is the value of `sqrt(1+(1)/(2^(2))+(1)/(3^(2)))+sqrt(1+(1)/(3^(2))+(1)/(4^(2)))+sqrt(1+(1)/(4^(2))+(1)/(5^(2)))` ?

Options

A. `(18)/(5)`
B. `(4)/(3)`
C. `(7)/(3)`
D. `(33)/(10)`

Answer

To solve the expression

\[ \sqrt{1 + \frac{1}{2^2} + \frac{1}{3^2}} + \sqrt{1 + \frac{1}{3^2} + \frac{1}{4^2}} + \sqrt{1 + \frac{1}{4^2} + \frac{1}{5^2}}, \]

we will evaluate each square root term step by step.

### Step 1: Simplify the first term

Calculate the first term:

\[ \sqrt{1 + \frac{1}{2^2} + \frac{1}{3^2}} = \sqrt{1 + \frac{1}{4} + \frac{1}{9}}. \]

To combine the fractions, we need a common denominator. The least common multiple of 4 and 9 is 36.

\[ 1 = \frac{36}{36}, \quad \frac{1}{4} = \frac{9}{36}, \quad \frac{1}{9} = \frac{4}{36}. \]

Now, add them together:

\[ \frac{36}{36} + \frac{9}{36} + \frac{4}{36} = \frac{49}{36}. \]

Thus, we have:

\[ \sqrt{1 + \frac{1}{2^2} + \frac{1}{3^2}} = \sqrt{\frac{49}{36}} = \frac{7}{6}. \]

### Step 2: Simplify the second term

Now calculate the second term:

\[ \sqrt{1 + \frac{1}{3^2} + \frac{1}{4^2}} = \sqrt{1 + \frac{1}{9} + \frac{1}{16}}. \]

The least common multiple of 9 and 16 is 144.

\[ 1 = \frac{144}{144}, \quad \frac{1}{9} = \frac{16}{144}, \quad \frac{1}{16} = \frac{9}{144}. \]

Now, add them together:

\[ \frac{144}{144} + \frac{16}{144} + \frac{9}{144} = \frac{169}{144}. \]

Thus, we have:

\[ \sqrt{1 + \frac{1}{3^2} + \frac{1}{4^2}} = \sqrt{\frac{169}{144}} = \frac{13}{12}. \]

### Step 3: Simplify the third term

Now calculate the third term:

\[ \sqrt{1 + \frac{1}{4^2} + \frac{1}{5^2}} = \sqrt{1 + \frac{1}{16} + \frac{1}{25}}. \]

The least common multiple of 16 and 25 is 400.

\[ 1 = \frac{400}{400}, \quad \frac{1}{16} = \frac{25}{400}, \quad \frac{1}{25} = \frac{16}{400}. \]

Now, add them together:

\[ \frac{400}{400} + \frac{25}{400} + \frac{16}{400} = \frac{441}{400}. \]

Thus, we have:

\[ \sqrt{1 + \frac{1}{4^2} + \frac{1}{5^2}} = \sqrt{\frac{441}{400}} = \frac{21}{20}. \]

### Step 4: Combine all terms

Now we can combine all three terms:

\[ \frac{7}{6} + \frac{13}{12} + \frac{21}{20}. \]

To add these fractions, we need a common denominator. The least common multiple of 6, 12, and 20 is 60.

Convert each fraction:

\[ \frac{7}{6} = \frac{70}{60}, \quad \frac{13}{12} = \frac{65}{60}, \quad \frac{21}{20} = \frac{63}{60}. \]

Now add them together:

\[ \frac{70}{60} + \frac{65}{60} + \frac{63}{60} = \frac{198}{60}. \]

### Step 5: Simplify the final result

Now simplify \(\frac{198}{60}\):

\[ \frac{198 \div 6}{60 \div 6} = \frac{33}{10}. \]

Thus, the final value is:

\[ \frac{33}{10} = 3.3. \]

### Final Answer

The value of the expression is \(\frac{33}{10}\) or \(3.3\).

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