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Chapter 7: Problem 78

Simplify. $$\frac{4-\sqrt{28}}{6}$$

Short Answer

Expert verified
\( \frac{2 - \sqrt{7}}{3} \)

Step by step solution

01

Simplify the Square Root

First, simplify the term under the square root. Notice that 28 can be expressed as the product of 4 and 7: \(\sqrt{28} = \sqrt{4 \cdot 7} \). Since \( \sqrt{4} = 2 \), we have: \( \sqrt{28} = 2\sqrt{7} \).
02

Substitute and Rewrite the Expression

Now substitute \( \sqrt{28} \) with \( 2\sqrt{7} \) in the original expression: \( \frac{4 - \sqrt{28}}{6} = \frac{4 - 2\sqrt{7}}{6} \).
03

Factor Out the Numerator

Factor out the greatest common factor in the numerator. The number 2 is a common factor: \( \frac{4 - 2\sqrt{7}}{6} = \frac{2(2 - \sqrt{7})}{6} \).
04

Simplify the Fraction

Simplify the fraction by dividing both the numerator and the denominator by 2: \( \frac{2(2 - \sqrt{7})}{6} = \frac{2 - \sqrt{7}}{3} \).

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Key Concepts

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

Square Root Simplification
Simplifying square roots involves breaking down the number under the root into its prime factors. This can make it easier to work with in equations. For example, take the square root of 28:
To simplify it, we look for perfect square factors in 28. Since 28 = 4 * 7, we can use this factorization:
\(\sqrt{28} = \sqrt{4 \cdot 7}\)
Knowing that \(\sqrt{4} = 2\), we can simplify further:
\(\sqrt{28} = 2 \sqrt{7}\)
This simplification helps because it turns a complex square root into the product of a whole number and a simpler square root. Practice identifying and breaking down factors to make square root simplification second nature.
Factoring Numerators
Factoring numerators is key when simplifying fractions that include square roots. It involves finding a common factor in the terms of the numerator. Let's look at our expression:
\(\frac{4 - 2 \sqrt{7}}{6}\)
Here, both 4 and \(2 \sqrt{7}\) share a common factor of 2. Factor out the 2:
\(\frac{4 - 2 \sqrt{7}}{6} = \frac{2(2 - \sqrt{7})}{6}\)
Factoring makes the expression easier to simplify further. Always look for common factors in the terms of your numerator to make fraction simplification steps straightforward.
Fraction Simplification
The final step in simplifying expressions with square roots often involves simplifying the fraction.
After factoring the numerator, divide the numerator and the denominator by the greatest common factor. In our example:
\(\frac{2(2 - \sqrt{7})}{6}\)
We know that both 2 and 6 can be divided by 2. So, simplify it as so:
\(\frac{2(2 - \sqrt{7})}{6} = \frac{2 - \sqrt{7}}{3}\)
This gives us our simplified fraction. By following these steps, you can tackle any fraction with square roots and simplify it efficiently. Make sure to always check if the numerator and denominator have common factors to reduce the fraction to its simplest form.

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