Problem 26 - Algebra 2 and Trigonometry

Question

In \(3-38\) write each expression in simplest form. Variables in the radicand with an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ 5 \sqrt{\frac{1}{5}}-\sqrt{\frac{1}{10}}+\sqrt{20} $$

Step-by-Step Solution

Answer

The expression simplifies to \( 3\sqrt{5} - \frac{\sqrt{10}}{10} \).

Step 1: Simplify Each Radical

First, simplify the radicals in the expression:1. \( \sqrt{\frac{1}{5}} = \frac{1}{\sqrt{5}} \). We rationalize the denominator by multiplying the numerator and the denominator by \( \sqrt{5} \) to get \( \frac{\sqrt{5}}{5} \).2. \( \sqrt{\frac{1}{10}} = \frac{1}{\sqrt{10}} \). Multiply by \( \sqrt{10} \) to rationalize: \( \frac{\sqrt{10}}{10} \).3. \( \sqrt{20} \) can be simplified as \( \sqrt{4 \times 5} = 2\sqrt{5} \).

Step 2: Rewrite the Expression

Rewrite the given expression after simplifying each part:\[ 5 \times \frac{\sqrt{5}}{5} - \frac{\sqrt{10}}{10} + 2\sqrt{5} \].

Step 3: Simplify the Expression

Combine like terms:1. The first term becomes \( \sqrt{5} \).2. The second term remains as \( -\frac{\sqrt{10}}{10} \).3. Combine \( \sqrt{5} + 2\sqrt{5} = 3\sqrt{5} \).The expression now is: \[ 3\sqrt{5} - \frac{\sqrt{10}}{10} \].

Step 4: Final Simplified Form

The expression in simplest form is: \[ 3\sqrt{5} - \frac{\sqrt{10}}{10} \].

Key Concepts

Simplifying Expressions

Simplifying expressions is about making them easier to work with by reducing them to their simplest form. This process involves eliminating any unnecessary complexities in the expression, such as complex fractions or radicals. For the given exercise, the idea is to reduce the expression \( 5 \sqrt{\frac{1}{5}} - \sqrt{\frac{1}{10}} + \sqrt{20} \) to something more manageable.Here are a few tips:

• Break Down Each Part: Focus on one part of the expression at a time. Look for numbers or variables that can be simplified.
• Combine Like Terms: Once simplified, combine like terms by adding or subtracting them as the case may be. This was done to combine \( \sqrt{5} + 2\sqrt{5} \) to get \( 3\sqrt{5} \).
• Look for Common Factors: In some cases, spotting common factors between terms can allow further simplification.

Simplifying expressions requires practice. The more you do it, the more familiar you'll become with spotting simplifications easily.

Rationalizing Denominators

Rationalizing the denominator involves removing any radical expressions in the denominator of a fraction. This is important because it transforms a fraction into a more standard mathematical form, which is often easier to work with and interpret.Here's how:

• Multiply Both Numerator and Denominator: Use the same radical that's in the denominator to multiply both the numerator and the denominator. For instance, \( \frac{1}{\sqrt{5}} \) is multiplied by \( \sqrt{5} \) to become \( \frac{\sqrt{5}}{5} \).
• Observe the Change: Notice how this process removes the radical from the denominator, making the expression look cleaner. \( \sqrt{10} \) was used similarly to rationalize \( \sqrt{\frac{1}{10}} \) to \( \frac{\sqrt{10}}{10} \).
• Simplify Further if Needed: After rationalizing, double-check the terms to see if they can be simplified again, either by combining like terms or breaking down any factors.

Understanding these steps is crucial for simplifying expressions involving radicals in measurement and algebra.

Radical Expressions

Radical expressions involve roots, such as square roots \( \sqrt{} \), cube roots, etc. These can sometimes make expressions look complex and intimidating. However, with a systematic approach, they can be simplified greatly.To simplify radical expressions effectively:

• Identify and Simplify: Learn to recognize squares (or cubes, etc.) within the radicand, the expression under the radical symbol. For example, \( \sqrt{20} \) can be broken down into \( \sqrt{4 \times 5} = 2\sqrt{5} \).
• Practice Prime Factorization: Breaking down numbers into their smallest prime components can reveal straightforward simplifications, as seen with radicals like \( \sqrt{20} \).
• Rationalize Consistently: Always apply rationalizing techniques to ensure the expression remains in its simplest rational form.

Remember, practice makes perfect. The more you work with radical expressions, the better you'll become at spotting potential simplifications.