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Chapter 10: Problem 101

Simplify completely. $$\frac{-10-\sqrt{50}}{5}$$

Short Answer

Expert verified
The simplified expression is \(-2 - \sqrt{2}\).

Step by step solution

01

Simplify the square root

In the numerator of the fraction, we have \(\sqrt{50}\). To simplify this square root, we have to find the prime factors of 50, and look for any pairs of the same number. The prime factors of 50 are 2 and 5, so we can rewrite \(\sqrt{50}\) as \(\sqrt{2 * 5^2}\). From this, we can extract the square root of any perfect squares in the expression. \[ \sqrt{50} = \sqrt{2 * 5^2} = 5\sqrt{2} \]
02

Rewrite the expression

Now that we have simplified the square root, we can substitute it back into the original expression. This gives us: \[ \frac{-10 - 5\sqrt{2}}{5} \]
03

Factor out the greatest common divisor (GCD)

Notice that both the numerator and the denominator of the fraction have 5 as their greatest common divisor. Therefore, we can factor out 5 in the numerator: \[ \frac{-10 - 5\sqrt{2}}{5} = \frac{5(-2 - \sqrt{2})}{5} \]
04

Simplify the fraction

Now, we can cancel out the common factor (5) in the numerator and the denominator: \[ \frac{5(-2 - \sqrt{2})}{5} = -2 - \sqrt{2} \] So the simplified expression is: \[ -2 - \sqrt{2} \]

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