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Chapter 9: Problem 324

In the following exercises, simplify. $$ \frac{10-\sqrt{200}}{20} $$

Short Answer

Expert verified
\( \frac{1 - \sqrt{2}}{2} \)

Step by step solution

01

Simplify the Square Root

First, simplify the square root in the numerator. Notice that \( \sqrt{200} \) can be written as \( \sqrt{100 \cdot 2} \). Since \( \sqrt{100} = 10 \), we have \( \sqrt{200} = 10 \sqrt{2} \).
02

Substitute Simplified Square Root

Substitute \( \sqrt{200} \) with \( 10 \sqrt{2} \) in the fraction. This gives us: \[ \frac{10 - 10 \sqrt{2}}{20} \]
03

Factor Out the Common Term

Factor out the common term in the numerator. This gives us: \[ \frac{10(1 - \sqrt{2})}{20} \]
04

Simplify the Fraction

Simplify the fraction by dividing both the numerator and the denominator by 10. This results in: \[ \frac{1 - \sqrt{2}}{2} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

simplifying square roots
To simplify algebraic expressions involving square roots, we need to break down the number inside the square root into its prime factors. For our exercise, we simplify \( \sqrt{200} \) by writing it in terms of its factors.
Break down \(200 \) into \(100 \cdot \2 \).
We know \( \sqrt{100} = 10 \), so \( \sqrt{200} = 10 \sqrt{2} \).
Using this method helps in expressing the square root in a simpler form, making it easier to handle in subsequent steps.
factoring algebraic expressions
Factoring is the process of breaking down an expression into a product of simpler expressions. In our exercise, once we simplified \( \sqrt{200} \) to \(10 \sqrt{2} \), we had: \[ \frac{10 - 10 \sqrt{2}}{20} \]
Next, we proceeded to factor out the common term \(10 \) in the numerator: \[ \frac{10(1 - \sqrt{2})}{20} \]
Factoring out common terms often helps to simplify fractions and makes the expression much more manageable.
fraction simplification
Simplifying fractions involves reducing the numerator and the denominator by their greatest common divisor. Considering our exercise:
We had: \[ \frac{10(1 - \sqrt{2})}{20} \]
Since \(10 \) is a common factor in both the numerator and the denominator, we divide both by \(10 \):
\[ \frac{10(1 - \sqrt{2})}{20} = \frac{1 - \sqrt{2}}{2} \]
This final simplification is much simpler to work with. Always look for common factors to make fractions simpler.
step-by-step problem solving
Solving complex problems becomes easier when broken down into smaller, manageable steps. For our exercise:
  • First, we simplified the square root \( \sqrt{200} \) to \(10 \sqrt{2} \).

  • Then, we substituted this back into the original fraction: \[ \frac{10 - 10 \sqrt{2}}{20} \]

  • Next, we factored out the common term \(10 \) in the numerator: \[ \frac{10(1 - \sqrt{2})}{20} \]

  • Finally, we simplified the fraction to get: \ \ \[ \frac{1 - \sqrt{2}}{2} \]

Each step builds on the previous, making the problem solving process clearer and much less daunting.

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