Problem 58 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \sqrt{\frac{16}{10}} $$

Step-by-Step Solution

Answer

The simplified expression of \(\sqrt{\frac{16}{10}}\) is \(\frac{2\sqrt{2}}{\sqrt{5}}\).

Step 1: Simplify the fraction

First, simplify the fraction \(\frac{16}{10}\) by dividing the numerator and the denominator by the greatest common divisor. In this case, it is 2. So, \(\frac{16}{10}\) simplifies to \(\frac{8}{5}\).

Step 2: Separate the fraction into two square roots

Next, use the property of square roots that \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\). Apply this property and separate \(\sqrt{\frac{8}{5}}\) into \(\frac{\sqrt{8}}{\sqrt{5}}\).

Step 3: Simplify the square roots

Then, simplify the square roots separately. The square root of 8 can be simplified to 2\(\sqrt{2}\), because 8 is equal to 4 * 2 and the square root of 4 is 2. So, \(\sqrt{8} = 2 * \sqrt{2}\). The square root of 5 can't be simplified and remains as \(\sqrt{5}\).

Step 4: Give the final answer

Finally, put the simplifications back into the fraction. The final simplified form of the expression \(\sqrt{\frac{16}{10}}\) is \(\frac{2\sqrt{2}}{\sqrt{5}}\).

Key Concepts

Greatest Common Divisor

To simplify any fraction, you first need to find the greatest common divisor (GCD). This is the largest number that can divide both the numerator and the denominator without leaving a remainder. Finding the GCD helps to reduce fractions to their simplest form. So, how do you find the GCD?

• List the factors of the numerator.
• List the factors of the denominator.
• Identify the greatest factor that appears in both lists.

For example, in the fraction \(\frac{16}{10}\), the factors of 16 are 1, 2, 4, 8, and 16, and the factors of 10 are 1, 2, 5, and 10. The greatest factor they share is 2. So, the GCD of 16 and 10 is 2. By dividing both the numerator and the denominator by their GCD, you simplify \(\frac{16}{10}\) to \(\frac{8}{5}\). This simplified fraction now makes further calculations easier.

Properties of Square Roots

Understanding the properties of square roots is crucial to simplifying expressions like \(\sqrt{\frac{a}{b}}\). One key property states: \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\). This means you can separate a square root of a fraction into individual square roots of the numerator and the denominator. This property is extremely helpful when simplifying complex square root expressions.
For example, \(\sqrt{\frac{8}{5}}\) can be rewritten as \(\frac{\sqrt{8}}{\sqrt{5}}\). This separation allows for the numerator and the denominator to be dealt with individually, and helps in finding simpler forms if factorization is possible. Remember, it's always good to check if the numbers inside any square roots can be broken down into factors, especially perfect squares. Perfect squares can be simplified further, making your calculations much easier.

Fraction Simplification

Simplifying fractions is an essential part of manipulating mathematical expressions. After separating square roots, each part might still need more simplification. Let's take an example from the given problem.

• Consider the numerator: \(\sqrt{8}\).
• Find factors, specifically looking for perfect squares: for 8, that's 4.
• Since 4 is a perfect square, \(\sqrt{8} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2} \).
• The denominator, \(\sqrt{5}\), is already in its simplest form because 5 is a prime number.

In conclusion, by performing such step-by-step simplifications, your final expression, \(\frac{2\sqrt{2}}{\sqrt{5}}\), becomes much clearer and manageable. Simplify separately, and then combine—this is the key to effective simplification of square roots and fractions.