The value of `sqrt(11+2sqrt(18))` is closest to :

Options

A. `4.8`
B. `4.4`
C. `3.8`
D. `4.1`

Answer

To solve the expression \( \sqrt{11 + 2\sqrt{18}} \), we can follow these steps:

### Step 1: Simplify the expression inside the square root
We start with the expression:
\[ \sqrt{11 + 2\sqrt{18}} \]
First, we need to simplify \( \sqrt{18} \):
\[ \sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2} \]
Now substitute this back into the expression:
\[ \sqrt{11 + 2(3\sqrt{2})} = \sqrt{11 + 6\sqrt{2}} \]

### Step 2: Assume the expression can be expressed as a binomial square
We assume that:
\[ \sqrt{11 + 6\sqrt{2}} = \sqrt{a} + \sqrt{b} \]
Squaring both sides gives:
\[ 11 + 6\sqrt{2} = a + b + 2\sqrt{ab} \]
From this, we can equate the rational and irrational parts:
1. \( a + b = 11 \)
2. \( 2\sqrt{ab} = 6\sqrt{2} \)

### Step 3: Solve for \( ab \)
From the second equation:
\[ \sqrt{ab} = 3\sqrt{2} \implies ab = 9 \cdot 2 = 18 \]

### Step 4: Set up a system of equations
Now we have a system of equations:
1. \( a + b = 11 \)
2. \( ab = 18 \)

### Step 5: Solve the quadratic equation
Let \( a \) and \( b \) be the roots of the quadratic equation:
\[ x^2 - (a+b)x + ab = 0 \implies x^2 - 11x + 18 = 0 \]
Using the quadratic formula:
\[ x = \frac{11 \pm \sqrt{11^2 - 4 \cdot 18}}{2} \]
Calculating the discriminant:
\[ 11^2 - 4 \cdot 18 = 121 - 72 = 49 \]
Thus:
\[ x = \frac{11 \pm 7}{2} \]
Calculating the two possible values:
1. \( x = \frac{18}{2} = 9 \)
2. \( x = \frac{4}{2} = 2 \)

### Step 6: Identify \( a \) and \( b \)
So we have \( a = 9 \) and \( b = 2 \).

### Step 7: Substitute back to find the square root
Now substituting back:
\[ \sqrt{11 + 6\sqrt{2}} = \sqrt{9} + \sqrt{2} = 3 + \sqrt{2} \]

### Step 8: Approximate the value of \( \sqrt{2} \)
Using the approximation \( \sqrt{2} \approx 1.414 \):
\[ 3 + \sqrt{2} \approx 3 + 1.414 = 4.414 \]

### Final Answer
Thus, the value of \( \sqrt{11 + 2\sqrt{18}} \) is closest to \( 4.4 \).

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