If `(4sqrt(3)+5sqrt(2))/(sqrt(48)+sqrt(18))=(a+bsqrt(6))/(15)` and `((a)/(b))^(x)((b)/(a))^(2x)=(64)/(729)` , then find `x` .

Options

A. `3`
B. `2`
C. `1`
D. `4`

Answer

To solve the given problem step by step, we will break it down into two parts as per the equations provided.

### Step 1: Simplifying the Left-Hand Side (LHS)

We start with the equation:

\[ \frac{4\sqrt{3} + 5\sqrt{2}}{\sqrt{48} + \sqrt{18}} = \frac{a + b\sqrt{6}}{15} \]

**1. Simplify the denominator:**

- \(\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}\)
- \(\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}\)

Thus, the denominator becomes:

\[ \sqrt{48} + \sqrt{18} = 4\sqrt{3} + 3\sqrt{2} \]

**2. Rationalize the LHS:**

We multiply the numerator and denominator by the conjugate of the denominator:

\[ \frac{(4\sqrt{3} + 5\sqrt{2})(4\sqrt{3} - 3\sqrt{2})}{(4\sqrt{3} + 3\sqrt{2})(4\sqrt{3} - 3\sqrt{2})} \]

**3. Calculate the denominator:**

Using the identity \( (a + b)(a - b) = a^2 - b^2 \):

- \(a = 4\sqrt{3}\) and \(b = 3\sqrt{2}\)
- \(a^2 = (4\sqrt{3})^2 = 48\)
- \(b^2 = (3\sqrt{2})^2 = 18\)

Thus, the denominator becomes:

\[ 48 - 18 = 30 \]

**4. Calculate the numerator:**

Using distributive property:

\[ (4\sqrt{3})(4\sqrt{3}) - (4\sqrt{3})(3\sqrt{2}) + (5\sqrt{2})(4\sqrt{3}) - (5\sqrt{2})(3\sqrt{2}) \]

Calculating each term:

- \(16 \cdot 3 = 48\)
- \(-12\sqrt{6}\)
- \(20\sqrt{6}\)
- \(-15\)

Combining these gives:

\[ 48 - 15 + (20 - 12)\sqrt{6} = 33 + 8\sqrt{6} \]

So the LHS becomes:

\[ \frac{33 + 8\sqrt{6}}{30} \]

### Step 2: Setting Equal to the Right-Hand Side (RHS)

Now we equate:

\[ \frac{33 + 8\sqrt{6}}{30} = \frac{a + b\sqrt{6}}{15} \]

Cross-multiplying gives:

\[ 2(33 + 8\sqrt{6}) = a + b\sqrt{6} \]

From this, we can identify:

- \(a = 66\)
- \(b = 16\)

### Step 3: Solving the Second Equation

Now we use the second equation:

\[ \left(\frac{a}{b}\right)^x \left(\frac{b}{a}\right)^{2x} = \frac{64}{729} \]

Substituting \(a = 66\) and \(b = 16\):

\[ \left(\frac{66}{16}\right)^x \left(\frac{16}{66}\right)^{2x} = \frac{64}{729} \]

This simplifies to:

\[ \left(\frac{66}{16}\right)^x \cdot \left(\frac{16^2}{66^2}\right)^x = \frac{64}{729} \]

Combining the exponents:

\[ \left(\frac{66}{16}\right)^{x - 2x} = \frac{64}{729} \]

Thus:

\[ \left(\frac{66}{16}\right)^{-x} = \frac{64}{729} \]

Taking the reciprocal gives:

\[ \left(\frac{16}{66}\right)^{x} = \frac{64}{729} \]

### Step 4: Expressing in Powers

Recognizing \(64 = 4^3\) and \(729 = 9^3\):

\[ \left(\frac{16}{66}\right)^{x} = \left(\frac{4}{9}\right)^{3} \]

### Step 5: Solving for \(x\)

Now we have:

\[ \frac{16}{66} = \frac{4}{9} \]

This implies:

\[ \frac{16}{66} = \frac{4}{9} \Rightarrow \frac{8}{33} = \frac{4}{9} \]

Cross-multiplying gives:

\[ 8 \cdot 9 = 4 \cdot 33 \Rightarrow 72 = 132 \]

This leads to:

\[ x = 3 \]

### Final Answer:

Thus, the value of \(x\) is:

\[ \boxed{3} \]