`(28-10sqrt(3))^(1//2)-(7+4sqrt(3))^(-1//2)` is equal to

Options

A. `4`
B. `5`
C. `3`
D. `4.3`

Answer

To solve the expression \((28 - 10\sqrt{3})^{1/2} - (7 + 4\sqrt{3})^{-1/2}\), we will break it down step by step.

### Step 1: Simplify the first term \((28 - 10\sqrt{3})^{1/2}\)

We can recognize that \(28 - 10\sqrt{3}\) can be expressed in a form that allows us to use the identity \(a^2 - 2ab + b^2 = (a - b)^2\).

Let’s assume:
- \(a^2 = 28\) and \(b^2 = 10\sqrt{3}\)

To find \(a\) and \(b\):
- \(a = \sqrt{28} = 2\sqrt{7}\)
- \(b = \sqrt{10\sqrt{3}} = \sqrt{10}\cdot\sqrt[4]{3}\)

Now, we can express \(28 - 10\sqrt{3}\) as:
\[ (2\sqrt{7} - \sqrt{10}\sqrt[4]{3})^2 \]

Thus,
\[ (28 - 10\sqrt{3})^{1/2} = 2\sqrt{7} - \sqrt{10}\sqrt[4]{3} \]

### Step 2: Simplify the second term \((7 + 4\sqrt{3})^{-1/2}\)

For the second term, we can express it as:
\[ (7 + 4\sqrt{3})^{-1/2} = \frac{1}{(7 + 4\sqrt{3})^{1/2}} \]

To simplify \((7 + 4\sqrt{3})^{1/2}\), we can use the same identity:
Assume:
- \(c^2 = 7\) and \(d^2 = 4\sqrt{3}\)

Finding \(c\) and \(d\):
- \(c = \sqrt{7}\)
- \(d = 2\sqrt[4]{3}\)

Thus,
\[ (7 + 4\sqrt{3})^{1/2} = \sqrt{7} + 2\sqrt[4]{3} \]

Therefore,
\[ (7 + 4\sqrt{3})^{-1/2} = \frac{1}{\sqrt{7} + 2\sqrt[4]{3}} \]

### Step 3: Combine the two terms

Now we can combine the two parts:
\[ (28 - 10\sqrt{3})^{1/2} - (7 + 4\sqrt{3})^{-1/2} = (2\sqrt{7} - \sqrt{10}\sqrt[4]{3}) - \frac{1}{\sqrt{7} + 2\sqrt[4]{3}} \]

### Step 4: Final simplification

To simplify further, we can find a common denominator and combine the terms. However, without loss of generality, we can evaluate the expression numerically or leave it in this form.

### Final Answer
The expression simplifies to:
\[ (2\sqrt{7} - \sqrt{10}\sqrt[4]{3}) - \frac{1}{\sqrt{7} + 2\sqrt[4]{3}} \]