√172 - √252

Steps to Solve

1. Simplifying the square roots

First, we need to simplify both square roots.

For $\sqrt{172}$, we can factor it:

$$ 172 = 4 \times 43 $$

So,

$$ \sqrt{172} = \sqrt{4 \times 43} = \sqrt{4} \times \sqrt{43} = 2\sqrt{43} $$

Next, for $\sqrt{252}$, we can factor it:

$$ 252 = 4 \times 63 $$

This gives us:

$$ \sqrt{252} = \sqrt{4 \times 63} = \sqrt{4} \times \sqrt{63} = 2\sqrt{63} $$

2. Further simplify $\sqrt{63}$

Now, we can simplify $\sqrt{63}$ further:

$$ 63 = 9 \times 7 $$

Thus,

$$ \sqrt{63} = \sqrt{9 \times 7} = \sqrt{9} \times \sqrt{7} = 3\sqrt{7} $$

3. Substituting the simplified values

Now substitute the simplified square roots back into the original expression:

$$ \sqrt{172} - \sqrt{252} = 2\sqrt{43} - 2\sqrt{63} = 2(\sqrt{43} - 3\sqrt{7}) $$

4. Final expression

So, the final simplified form of the expression is:

$$ 2(\sqrt{43} - 3\sqrt{7}) $$