Question

Combine the radicals using addition and subtraction, if possible.

548−−√−63–√+212

Answer 1

the answer is a square root

Answer 2

To combine the radicals \( \sqrt{548} - \sqrt{63} + 212 \), we first simplify the square roots if possible.

1. **Simplify \( \sqrt{548} \)**:
\[
\sqrt{548} = \sqrt{4 \times 137} = \sqrt{4} \times \sqrt{137} = 2\sqrt{137}
\]

2. **Simplify \( \sqrt{63} \)**:
\[
\sqrt{63} = \sqrt{9 \times 7} = \sqrt{9} \times \sqrt{7} = 3\sqrt{7}
\]

3. **Now substitute these simplified radicals back into the expression**:
\[
2\sqrt{137} - 3\sqrt{7} + 212
\]

Since \( 2\sqrt{137} \) and \( 3\sqrt{7} \) are not like radicals, they cannot be combined further.

So, the final expression is:
\[
2\sqrt{137} - 3\sqrt{7} + 212
\]

Answer 3

Apologies for not addressing the request properly in my previous response. Let's take another look at the expression and see how we can combine \( \sqrt{548} - \sqrt{63} + 212 \) to express it as a single square root if possible.

1. **Calculate \( \sqrt{548} \)**:
\[
\sqrt{548} = \sqrt{4 \cdot 137} = 2\sqrt{137}
\]

2. **Calculate \( \sqrt{63} \)**:
\[
\sqrt{63} = \sqrt{9 \cdot 7} = 3\sqrt{7}
\]

Now, substituting these values into the expression gives us:
\[
2\sqrt{137} - 3\sqrt{7} + 212
\]

To express this as a single square root, we can examine the possibility of combining the terms linearly or regrouping them in terms of a relevant expression. However, since \( 212 \) does not combine directly with the radical terms neatly, we seek to represent this expression as an equivalent square root.

Let's break down the expression:
\[
2\sqrt{137} - 3\sqrt{7} + 212
\]

To put it into a single square root, we might have:
\[
\text{Let } x = 2\sqrt{137} - 3\sqrt{7} + 212
\]

But determining it directly as one square root is complex without additional specified forms. To simplify this further, we can calculate:
- Calculate approximate values:
- \( \sqrt{137} \approx 11.7 \), thus \( 2\sqrt{137} \approx 23.4 \)
- \( \sqrt{7} \approx 2.645 \), thus \( 3\sqrt{7} \approx 7.935 \)

This gives us an approximate value as:
\[
23.4 - 7.935 + 212 \approx 227.465
\]

Therefore, if interpreting back to a square root is required, we would typically aim for simplifications. However, \( 2\sqrt{137} - 3\sqrt{7} + 212 \) might not be factored easily into a singular square-root format without specific goals or expressions involved.

If the question intends for further algebraic manipulation or specific value checks, please clarify or adjust your request to fit a certain condition. Thank you!