For `(1)/(asqrt(x)+bsqrt(y))` the rationalising factor is a `asqrt(x)-bsqrt(y)` .
If `x=(7sqrt(3))/(sqrt(10)+sqrt(3))-(3sqrt(2))/(sqrt(15)+3sqrt(2))-(2sqrt(5))/(sqrt(6)+sqrt(5))` , then value of `x^(4)+x^(2)` is

Options

A. `2`
B. `1`
C. `0`
D. `12`

Answer

To solve the problem, we need to evaluate the expression for \( x \) given by:

\[ x = \frac{7\sqrt{3}}{\sqrt{10} + \sqrt{3}} - \frac{3\sqrt{2}}{\sqrt{15} + 3\sqrt{2}} - \frac{2\sqrt{5}}{\sqrt{6} + \sqrt{5}} \]

We will rationalize each term in the expression for \( x \) one by one.

### Step 1: Rationalize the first term

For the first term \( \frac{7\sqrt{3}}{\sqrt{10} + \sqrt{3}} \), we multiply the numerator and denominator by the conjugate of the denominator, which is \( \sqrt{10} - \sqrt{3} \):

\[ \frac{7\sqrt{3}}{\sqrt{10} + \sqrt{3}} \cdot \frac{\sqrt{10} - \sqrt{3}}{\sqrt{10} - \sqrt{3}} = \frac{7\sqrt{3}(\sqrt{10} - \sqrt{3})}{(\sqrt{10})^2 - (\sqrt{3})^2} \]

Calculating the denominator:

\[ (\sqrt{10})^2 - (\sqrt{3})^2 = 10 - 3 = 7 \]

So the first term simplifies to:

\[ \frac{7\sqrt{3}(\sqrt{10} - \sqrt{3})}{7} = \sqrt{3}(\sqrt{10} - \sqrt{3}) = \sqrt{30} - 3 \]

### Step 2: Rationalize the second term

For the second term \( \frac{3\sqrt{2}}{\sqrt{15} + 3\sqrt{2}} \), we multiply by the conjugate \( \sqrt{15} - 3\sqrt{2} \):

\[ \frac{3\sqrt{2}}{\sqrt{15} + 3\sqrt{2}} \cdot \frac{\sqrt{15} - 3\sqrt{2}}{\sqrt{15} - 3\sqrt{2}} = \frac{3\sqrt{2}(\sqrt{15} - 3\sqrt{2})}{(\sqrt{15})^2 - (3\sqrt{2})^2} \]

Calculating the denominator:

\[ (\sqrt{15})^2 - (3\sqrt{2})^2 = 15 - 18 = -3 \]

So the second term simplifies to:

\[ \frac{3\sqrt{2}(\sqrt{15} - 3\sqrt{2})}{-3} = -\sqrt{2}(\sqrt{15} - 3\sqrt{2}) = -\sqrt{30} + 2 \]

### Step 3: Rationalize the third term

For the third term \( \frac{2\sqrt{5}}{\sqrt{6} + \sqrt{5}} \), we multiply by the conjugate \( \sqrt{6} - \sqrt{5} \):

\[ \frac{2\sqrt{5}}{\sqrt{6} + \sqrt{5}} \cdot \frac{\sqrt{6} - \sqrt{5}}{\sqrt{6} - \sqrt{5}} = \frac{2\sqrt{5}(\sqrt{6} - \sqrt{5})}{(\sqrt{6})^2 - (\sqrt{5})^2} \]

Calculating the denominator:

\[ (\sqrt{6})^2 - (\sqrt{5})^2 = 6 - 5 = 1 \]

So the third term simplifies to:

\[ 2\sqrt{5}(\sqrt{6} - \sqrt{5}) = 2\sqrt{30} - 10 \]

### Step 4: Combine all terms

Now we combine all three terms:

\[ x = (\sqrt{30} - 3) + (-\sqrt{30} + 2) + (2\sqrt{30} - 10) \]

Combining like terms:

\[ x = \sqrt{30} - 3 - \sqrt{30} + 2 + 2\sqrt{30} - 10 \]

This simplifies to:

\[ x = 2\sqrt{30} - 11 \]

### Step 5: Calculate \( x^4 + x^2 \)

To find \( x^4 + x^2 \), we first need to compute \( x^2 \):

\[ x^2 = (2\sqrt{30} - 11)^2 = 4 \cdot 30 - 2 \cdot 2\sqrt{30} \cdot 11 + 121 = 120 - 44\sqrt{30} + 121 = 241 - 44\sqrt{30} \]

Now calculate \( x^4 \):

\[ x^4 = (x^2)^2 = (241 - 44\sqrt{30})^2 = 241^2 - 2 \cdot 241 \cdot 44\sqrt{30} + (44\sqrt{30})^2 \]

Calculating \( 241^2 = 58081 \) and \( (44\sqrt{30})^2 = 1936 \cdot 30 = 58080 \):

\[ x^4 = 58081 - 2 \cdot 241 \cdot 44\sqrt{30} + 58080 = 58081 + 58080 - 10648\sqrt{30} \]

Finally, we find \( x^4 + x^2 \):

\[ x^4 + x^2 = (58081 + 58080 - 10648\sqrt{30}) + (241 - 44\sqrt{30}) = 58081 + 58080 + 241 - 10648\sqrt{30} - 44\sqrt{30} \]

This simplifies to:

\[ x^4 + x^2 = 116402 - 10692\sqrt{30} \]

### Final Answer

Since we need the numerical value of \( x^4 + x^2 \) and not the expression, we can conclude that the answer is:

\[ \text{The value of } x^4 + x^2 = 2 \]