Problem 50 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \sqrt{\frac{3}{5}} $$

Step-by-Step Solution

Answer

\(\frac{\sqrt{15}}{5}\)

Step 1: Recognize the Structure

Identify the problem as a task to simplify the square root of a fraction. Therefore, the square root will need to be applied to both the numerator and the denominator separately because \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\). Our fraction is thus \(\frac{3}{5}\), so \(a=3\) and \(b=5\).

Step 2: Apply the Square Root

Apply the square root separately to the numerator and the denominator to get \(\frac{\sqrt{3}}{\sqrt{5}}\).

Step 3: Rationalize the Denominator

Rationalize the denominator, which means to eliminate the square root from the denominator. To do this, multiply both the numerator and denominator by the denominator's square root, which in this case is \(\sqrt{5}\). So the multiplication gives \(\frac{\sqrt{3} \times \sqrt{5}}{\sqrt{5} \times \sqrt{5}}\).

Step 4: Simplify the Expression

Now, simplifying the expression we obtain \(\frac{\sqrt{15}}{5}\).

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator is a technique used in algebra to eliminate square roots from the denominator of a fraction. This is achieved by multiplying both the numerator and the denominator by the same square root that is in the denominator. For example, if you have a denominator of \( \sqrt{b} \), you would multiply the entire fraction by \( \frac{\sqrt{b}}{\sqrt{b}} \). This transforms the denominator into a rational number because \( \sqrt{b} \times \sqrt{b} = b \).
This process helps simplify expressions and is often necessary for further calculations. It's a key step in making expressions easier to work with in algebra.

Square Root of a Fraction

Finding the square root of a fraction involves applying the square root separately to both the numerator and the denominator. If you have a fraction \( \frac{a}{b} \), the square root of the fraction is \( \frac{\sqrt{a}}{\sqrt{b}} \).

• This step allows you to simplify the expression in parts.
• It's the initial breakdown before you move on to rationalizing the denominator or further simplifications.

Understanding this concept helps you tackle more complex algebraic expressions with ease.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They can represent complex relationships and are foundational in mathematics. In this context, expressions involving square roots and fractions require manipulation using basic algebraic rules.
Learning to skillfully navigate algebraic expressions is crucial for simplifying, solving, and understanding more advanced mathematical concepts. It's all about understanding how these elements work together to form an expression.

Basic Algebraic Techniques

Basic algebraic techniques are the strategies and rules used to manipulate algebraic expressions. These include simplifying terms, expanding expressions, and rationalizing denominators.

• Simplifying an expression means reducing it to its simplest form without changing its value.
• Rationalizing the denominator, as previously mentioned, eliminates irrational numbers from the denominator, making calculations simpler.

Mastering these techniques allows for effective problem-solving and lays the groundwork for more advanced algebra and calculus.