Question

Simplify and rational this:
(3 + root 8)/ (2 + root 2)

Answer 1

Can you the expanding of the numerator again

Answer 2

To simplify and rationalize the expression \(\frac{3 + \sqrt{8}}{2 + \sqrt{2}}\), follow these steps:

1. **Simplify \(\sqrt{8}\)**:
\[
\sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4}\sqrt{2} = 2\sqrt{2}
\]
So, replace \(\sqrt{8}\) in the original expression:
\[
\frac{3 + 2\sqrt{2}}{2 + \sqrt{2}}
\]

2. **Multiply by the conjugate of the denominator**:
The conjugate of \(2 + \sqrt{2}\) is \(2 - \sqrt{2}\). Therefore, we multiply both the numerator and the denominator by this conjugate:
\[
\frac{(3 + 2\sqrt{2})(2 - \sqrt{2})}{(2 + \sqrt{2})(2 - \sqrt{2})}
\]

3. **Simplify the denominator**:
\[
(2 + \sqrt{2})(2 - \sqrt{2}) = 2^2 - (\sqrt{2})^2 = 4 - 2 = 2
\]

4. **Expand the numerator**:
\[
(3 + 2\sqrt{2})(2 - \sqrt{2}) = 3 \cdot 2 + 3 \cdot (-\sqrt{2}) + 2\sqrt{2} \cdot 2 + 2\sqrt{2} \cdot (-\sqrt{2})
\]
This simplifies to:
\[
= 6 - 3\sqrt{2} + 4\sqrt{2} - 2 = 4 + \sqrt{2}
\]

5. **Combine both parts**:
Thus the expression becomes:
\[
\frac{4 + \sqrt{2}}{2}
\]

6. **Simplify further**:
Divide each term in the numerator by 2:
\[
= \frac{4}{2} + \frac{\sqrt{2}}{2} = 2 + \frac{\sqrt{2}}{2}
\]

The final simplified and rationalized expression is:
\[
2 + \frac{\sqrt{2}}{2}
\]

Answer 3

Sure! Let's carefully expand the numerator of the expression \((3 + 2\sqrt{2})(2 - \sqrt{2})\) step by step.

We are multiplying:

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

To expand this, we use the distributive property (also known as the FOIL method for binomials):

1. **First**: Multiply the first terms:
\[
3 \cdot 2 = 6
\]

2. **Outer**: Multiply the outer terms:
\[
3 \cdot (-\sqrt{2}) = -3\sqrt{2}
\]

3. **Inner**: Multiply the inner terms:
\[
2\sqrt{2} \cdot 2 = 4\sqrt{2}
\]

4. **Last**: Multiply the last terms:
\[
2\sqrt{2} \cdot (-\sqrt{2}) = -2(\sqrt{2})^2 = -2 \cdot 2 = -4
\]

Now, let's combine these results:

\[
6 - 3\sqrt{2} + 4\sqrt{2} - 4
\]

Combine the like terms:

- The constant terms: \(6 - 4 = 2\)
- The \(\sqrt{2}\) terms: \(-3\sqrt{2} + 4\sqrt{2} = 1\sqrt{2}\) or simply \(\sqrt{2}\)

Putting it all together, we have:

\[
2 + \sqrt{2}
\]

So the expanded numerator is:

\[
2 + \sqrt{2}
\]

Thus, combining everything, the entire expression becomes:

\[
\frac{2 + \sqrt{2}}{2}
\]

Which simplifies to:

\[
1 + \frac{\sqrt{2}}{2}
\]

So the final simplified and rationalized result is:

\[
1 + \frac{\sqrt{2}}{2}
\]

Thank you for your patience!