If `((8+3sqrt(2))+(7-sqrt(2))-(3-4sqrt(2)))/(6-2sqrt(2))=a+bsqrt(2)`, then find the value of a and b respectively.

Options

A. `(24)/(7),(15sqrt(2))/(7)`
B. `(-24)/(7),(-15)/(7)`
C. `(24)/(7),(15)/(6)`
D. `(24)/(7),(15)/(7)`

Answer

To solve the equation

\[ \frac{(8 + 3\sqrt{2}) + (7 - \sqrt{2}) - (3 - 4\sqrt{2})}{6 - 2\sqrt{2}} = a + b\sqrt{2}, \]

we will simplify the numerator and denominator step by step.

### Step 1: Simplify the Numerator

First, we simplify the expression in the numerator:

\[ (8 + 3\sqrt{2}) + (7 - \sqrt{2}) - (3 - 4\sqrt{2}). \]

Distributing the negative sign in the last term gives:

\[ (8 + 3\sqrt{2}) + (7 - \sqrt{2}) - 3 + 4\sqrt{2}. \]

Now, combine like terms:

- The constant terms: \(8 + 7 - 3 = 12\).
- The \(\sqrt{2}\) terms: \(3\sqrt{2} - \sqrt{2} + 4\sqrt{2} = (3 - 1 + 4)\sqrt{2} = 6\sqrt{2}\).

Thus, the numerator simplifies to:

\[ 12 + 6\sqrt{2}. \]

### Step 2: Simplify the Denominator

Next, we simplify the denominator:

\[ 6 - 2\sqrt{2}. \]

### Step 3: Write the Expression

Now, we can rewrite the entire expression:

\[ \frac{12 + 6\sqrt{2}}{6 - 2\sqrt{2}}. \]

### Step 4: Rationalize the Denominator

To simplify this expression further, we will multiply the numerator and denominator by the conjugate of the denominator, which is \(6 + 2\sqrt{2}\):

\[ \frac{(12 + 6\sqrt{2})(6 + 2\sqrt{2})}{(6 - 2\sqrt{2})(6 + 2\sqrt{2})}. \]

Calculating the denominator first:

\[ (6 - 2\sqrt{2})(6 + 2\sqrt{2}) = 6^2 - (2\sqrt{2})^2 = 36 - 8 = 28. \]

Now, calculating the numerator:

\[ (12 + 6\sqrt{2})(6 + 2\sqrt{2}) = 12 \cdot 6 + 12 \cdot 2\sqrt{2} + 6\sqrt{2} \cdot 6 + 6\sqrt{2} \cdot 2\sqrt{2}. \]

Calculating each term:

- \(12 \cdot 6 = 72\),
- \(12 \cdot 2\sqrt{2} = 24\sqrt{2}\),
- \(6\sqrt{2} \cdot 6 = 36\sqrt{2}\),
- \(6\sqrt{2} \cdot 2\sqrt{2} = 12 \cdot 2 = 24\).

Combining these gives:

\[ 72 + (24\sqrt{2} + 36\sqrt{2}) + 24 = 96 + 60\sqrt{2}. \]

### Step 5: Final Expression

Now we have:

\[ \frac{96 + 60\sqrt{2}}{28}. \]

We can simplify this by dividing each term by 28:

\[ \frac{96}{28} + \frac{60\sqrt{2}}{28} = \frac{24}{7} + \frac{15\sqrt{2}}{7}. \]

### Conclusion

Thus, we can express this as:

\[ a + b\sqrt{2} \quad \text{where} \quad a = \frac{24}{7}, \, b = \frac{15}{7}. \]

### Final Values

So, the values of \(a\) and \(b\) are:

\[ a = \frac{24}{7}, \quad b = \frac{15}{7}. \]