To evaluate the expression \(-\sqrt{424} + \sqrt{12} + \sqrt{56} + \sqrt{427}\), we need to compute the square roots and then combine the results.
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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.
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Rewrite the expression with the simplifications: \[ -\sqrt{424} + \sqrt{12} + \sqrt{56} + \sqrt{427} = -2\sqrt{106} + 2\sqrt{3} + 2\sqrt{14} + \sqrt{427} \]
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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:
- \(-2\sqrt{106}\)
- \(2\sqrt{3}\)
- \(2\sqrt{14}\)
- \(\sqrt{427}\)
And the approximate numerical result is \(10.995\).