Let's evaluate the expression \(\sqrt{28 + 10\sqrt{3}} - \sqrt{7 + 4\sqrt{3}}\).

### Step 1: Simplify each square root

Assume:
\[
\sqrt{28 + 10\sqrt{3}} = \sqrt{a} + \sqrt{b}
\]
where \( a \) and \( b \) are to be determined.

Squaring both sides:
\[
28 + 10\sqrt{3} = a + b + 2\sqrt{ab}
\]
Matching the rational and irrational parts, we get:
\[
a + b = 28 \quad \text{(1)}
\]
\[
2\sqrt{ab} = 10\sqrt{3} \quad \Rightarrow \quad ab = 75 \quad \text{(2)}
\]

We need to solve the quadratic equation formed by \( x^2 - (a+b)x + ab = 0 \):
\[
x^2 - 28x + 75 = 0
\]
The discriminant is:
\[
\Delta = 28^2 - 4 \times 75 = 784 - 300 = 484
\]
The square root of 484 is 22, so:
\[
x = \frac{28 \pm 22}{2}
\]
Thus, \( x = 25 \) or \( x = 3 \).

Therefore, \( a = 25 \) and \( b = 3 \), or vice versa. Thus:
\[
\sqrt{28 + 10\sqrt{3}} = \sqrt{25} + \sqrt{3} = 5 + \sqrt{3}
\]

Similarly, for \(\sqrt{7 + 4\sqrt{3}}\), assume:
\[
\sqrt{7 + 4\sqrt{3}} = \sqrt{c} + \sqrt{d}
\]
Squaring both sides:
\[
7 + 4\sqrt{3} = c + d + 2\sqrt{cd}
\]
We get:
\[
c + d = 7 \quad \text{(3)}
\]
\[
2\sqrt{cd} = 4\sqrt{3} \quad \Rightarrow \quad cd = 12 \quad \text{(4)}
\]

The quadratic equation is:
\[
x^2 - 7x + 12 = 0
\]
The discriminant is:
\[
\Delta = 7^2 - 4 \times 12 = 49 - 48 = 1
\]
So:
\[
x = \frac{7 \pm 1}{2}
\]
Thus, \( x = 6 \) or \( x = 1 \).

Therefore, \( c = 6 \) and \( d = 1 \), or vice versa. Thus:
\[
\sqrt{7 + 4\sqrt{3}} = \sqrt{6} + \sqrt{1} = \sqrt{6} + 1
\]

### Step 2: Subtract the two square roots

Subtracting:
\[
\sqrt{28 + 10\sqrt{3}} - \sqrt{7 + 4\sqrt{3}} = (5 + \sqrt{3}) - (2 + \sqrt{3}) = 3
\]

So the expression simplifies to:
\(\boxed{3}\).

Would you like more details or have any questions?

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Here are 5 related questions you might be interested in:

1. How do you simplify expressions involving square roots?
2. What is the general method to simplify square roots of the form \( \sqrt{a + b\sqrt{c}} \)?
3. Can every expression of the form \( \sqrt{a + b\sqrt{c}} \) be simplified in this way?
4. How do you solve quadratic equations with square roots?
5. What are other methods to approach square root expressions?

**Tip:** When simplifying square roots of expressions like \( \sqrt{a + b\sqrt{c}} \), checking if the expression can be expressed as \( \sqrt{x} + \sqrt{y} \) can simplify the process greatly!

Learn how to evaluate the expression sqrt(28 + 10sqrt(3)) - sqrt(7 + 4sqrt(3)) step-by-step. This problem involves simplifying square roots and applying the quadratic equation formula. Suitable for advanced high school students.

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