Question

What is −424−−√+12−−√+56–√+427−−√
?

Enter the missing values in the solution.

(1 point)
$$

Answer 1

To evaluate the expression \(-\sqrt{424} + \sqrt{12} + \sqrt{56} + \sqrt{427}\), we need to compute the square roots and then combine the results.

1. **Calculate each square root**:
- \(\sqrt{424} = \sqrt{4 \times 106} = 2\sqrt{106}\)
- \(\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}\)
- \(\sqrt{56} = \sqrt{4 \times 14} = 2\sqrt{14}\)
- \(\sqrt{427}\) is already in its simplest form.

2. **Rewrite the expression with the simplifications**:
\[
-\sqrt{424} + \sqrt{12} + \sqrt{56} + \sqrt{427} = -2\sqrt{106} + 2\sqrt{3} + 2\sqrt{14} + \sqrt{427}
\]

3. **Combine like terms**:
\[
= 2\sqrt{3} + 2\sqrt{14} - 2\sqrt{106} + \sqrt{427}
\]

Since the values cannot be further simplified without numerical evaluation, the expression can be rearranged to emphasize the components:

\[
2\sqrt{3} + 2\sqrt{14} - 2\sqrt{106} + \sqrt{427}
\]

If you want to approximate the square roots numerically:

- \(\sqrt{424} \approx 20.616\)
- \(\sqrt{12} \approx 3.464\)
- \(\sqrt{56} \approx 7.483\)
- \(\sqrt{427} \approx 20.674\)

Now putting the approximate values into the expression:

\[
-\sqrt{424} + \sqrt{12} + \sqrt{56} + \sqrt{427} \approx -20.616 + 3.464 + 7.483 + 20.674
\]

After calculating this step-by-step:

\(-20.616 + 3.464 = -17.152\)

\(-17.152 + 7.483 \approx -9.669\)

Finally,

\(-9.669 + 20.674 \approx 10.995\)

Thus, the final approximate value of the expression is roughly \(10.995\).

Fill in the missing values as:

1. \(-2\sqrt{106}\)
2. \(2\sqrt{3}\)
3. \(2\sqrt{14}\)
4. \(\sqrt{427}\)

And the approximate numerical result is \(10.995\).