\left(\sqrt{10}+\sqrt{6}\right)\sqrt{8-2\sqrt{15}}

We need to simplify the expression:

\[
\left(\sqrt{10} + \sqrt{6}\right) \sqrt{8 - 2\sqrt{15}}
\]

### Step 1: Simplify the expression under the square root
We first focus on simplifying \(\sqrt{8 - 2\sqrt{15}}\). This expression is of the form \(\sqrt{a - 2\sqrt{b}}\), which can sometimes be expressed as:

\[
\sqrt{a - 2\sqrt{b}} = \sqrt{x} - \sqrt{y}
\]

To apply this, we need \(a = x + y\) and \(b = xy\).

#### Finding \(x\) and \(y\):
- Here \(a = 8\) and \(b = 15\).
- We want to find \(x\) and \(y\) such that:
  \[
  x + y = 8 \quad \text{and} \quad xy = 15
  \]

These are the two equations we need to solve. Let’s solve them by factoring.

From the equation \(x + y = 8\) and \(xy = 15\), \(x\) and \(y\) must be the solutions to the quadratic equation:

\[
t^2 - (x + y)t + xy = 0
\]

Substituting \(x + y = 8\) and \(xy = 15\), we get:

\[
t^2 - 8t + 15 = 0
\]

Solving this quadratic equation using the quadratic formula:

\[
t = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(1)(15)}}{2(1)}
\]

\[
t = \frac{8 \pm \sqrt{64 - 60}}{2} = \frac{8 \pm \sqrt{4}}{2} = \frac{8 \pm 2}{2}
\]

Thus, \(t = 5\) and \(t = 3\), so \(x = 5\) and \(y = 3\).

### Step 2: Substitute and simplify
Now that we have \(x = 5\) and \(y = 3\), we can rewrite the square root as:

\[
\sqrt{8 - 2\sqrt{15}} = \sqrt{5} - \sqrt{3}
\]

### Step 3: Expand the original expression
We now substitute this into the original expression:

\[
\left(\sqrt{10} + \sqrt{6}\right)(\sqrt{5} - \sqrt{3})
\]

Now, expand using the distributive property:

\[
= \sqrt{10} \cdot \sqrt{5} - \sqrt{10} \cdot \sqrt{3} + \sqrt{6} \cdot \sqrt{5} - \sqrt{6} \cdot \sqrt{3}
\]

Simplifying each term:

\[
= \sqrt{50} - \sqrt{30} + \sqrt{30} - \sqrt{18}
\]

\[
= 5\sqrt{2} - \sqrt{18}
\]

### Step 4: Simplify \(\sqrt{18}\)
We can simplify \(\sqrt{18}\) as:

\[
\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}
\]

### Step 5: Final simplification
Substitute this back:

\[
5\sqrt{2} - 3\sqrt{2} = 2\sqrt{2}
\]

Thus, the simplified expression is:

\[
\boxed{2\sqrt{2}}
\]

Let me know if you have any questions or if you would like further details!

---

Here are some related questions to practice:
1. Simplify \(\sqrt{12 - 2\sqrt{35}}\).
2. Expand \((\sqrt{7} + \sqrt{5})(\sqrt{3} + \sqrt{2})\).
3. Solve \(\sqrt{5 + 2\sqrt{6}}\) in its simplified form.
4. Simplify \(\sqrt{6 + 2\sqrt{8}}\).
5. Expand \((\sqrt{8} - \sqrt{6})(\sqrt{10} + \sqrt{2})\).

**Tip**: When simplifying expressions involving square roots, look for patterns of the form \(\sqrt{a \pm 2\sqrt{b}}\), as they often simplify nicely.

\left(\sqrt{10} + \sqrt{6}\right)\sqrt{8 - 2\sqrt{15}}

Learn how to simplify the expression \left(\sqrt{10} + \sqrt{6}\right)\sqrt{8 - 2\sqrt{15}} with this detailed step-by-step solution. The problem involves simplifying radicals, using the quadratic formula, and applying the distributive property. Suitable for students in Grades 9-11.

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