if it is known that a is 2 and b is 2 the square root of 2 then the result is (squared root)....... (using Pythagoras)

Given that \(a = 2\) and \(b = 2\sqrt{2}\), we can use the Pythagorean theorem to find the hypotenuse \(c\) of a right triangle with sides \(a\) and \(b\).

The Pythagorean theorem states:

\[
c = \sqrt{a^2 + b^2}
\]

Substitute the values of \(a\) and \(b\) into the formula:

\[
c = \sqrt{2^2 + (2\sqrt{2})^2}
\]

Now, calculate each term inside the square root:

1. \(2^2 = 4\)
2. \((2\sqrt{2})^2 = 2^2 \cdot (\sqrt{2})^2 = 4 \cdot 2 = 8\)

Add these results together:

\[
c = \sqrt{4 + 8}
\]

\[
c = \sqrt{12}
\]

The square root of 12 can be simplified further:

\[
c = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}
\]

So, the length of the hypotenuse \(c\) is \(2\sqrt{3}\).

### Summary
Given \(a = 2\) and \(b = 2\sqrt{2}\), using the Pythagorean theorem, the result for the hypotenuse \(c\) is \(2\sqrt{3}\).

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### Related Questions
1. What is the Pythagorean theorem, and how is it derived?
2. How do you simplify square roots like \(\sqrt{12}\)?
3. What are the properties of right triangles in relation to the Pythagorean theorem?
4. Can you explain the process of squaring numbers involving square roots?
5. What are some real-world applications of the Pythagorean theorem?

### Tip
When dealing with square roots, always look for ways to simplify by finding perfect squares within the radical.

Explore how to find the hypotenuse using the Pythagorean theorem with specific values of a = 2 and b = 2√2. This problem involves applying the Pythagorean theorem to a right triangle, calculating square roots, and simplifying radicals. Suitable for high school students.

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