Problem 75 - Algebra 2

Question

Simplify each expression. \(\frac{3}{\sqrt{5}}\)

Step-by-Step Solution

Answer

\( \frac{3\sqrt{5}}{5} \)

Step 1: Identify the Problem

The given expression is \( \frac{3}{\sqrt{5}} \). Our task is to simplify this expression by eliminating the square root from the denominator.

Step 2: Rationalize the Denominator

To remove the square root from the denominator, we must multiply both the numerator and the denominator by \( \sqrt{5} \), which will not change the value of the expression but will eliminate the radical from the denominator.

Step 3: Multiply Numerator and Denominator

Multiply the numerator by \( \sqrt{5} \), resulting in \( 3 \times \sqrt{5} = 3\sqrt{5} \). Similarly, multiply the denominator by \( \sqrt{5} \), resulting in \( \sqrt{5} \times \sqrt{5} = 5 \).

Step 4: Write the Simplified Expression

Having multiplied both parts by \( \sqrt{5} \), the fraction becomes \( \frac{3\sqrt{5}}{5} \), which is the simplified form.

Key Concepts

Simplifying Expressions

Simplifying expressions is a crucial skill in algebra that helps make complex expressions more manageable and easier to interpret. When simplifying an expression, the main goal is to rewrite it in the simplest form possible without changing its value. This often involves

• combining like terms,
• canceling out common factors,
• or, as in our exercise, rationalizing the denominator.

In the given expression \(\frac{3}{\sqrt{5}}\), simplifying involves eliminating the square root from the denominator. This process ensures that the expression is mathematically acceptable and easier to compute.

Square Roots

Square roots are fundamental in mathematics, representing the number that, when multiplied by itself, gives the original number. For instance, the square root of 25 is 5 because \(5 \times 5 = 25\). In many mathematical scenarios, particularly in algebra, it's necessary to manipulate square roots. This might include rationalizing denominators, where a square root appears in the denominator of a fraction. This makes the expression simpler in conventional terms, as displayed by rationalizing \(\sqrt{5}\) in \(\frac{3}{\sqrt{5}}\). By multiplying both the numerator and the denominator by \(\sqrt{5}\), the denominator becomes a whole number, thereby simplifying the expression.

Fractions

Fractions represent the division of one quantity by another, presented as one number over another, separated by a line. In algebra, fractions can become complex, especially when they involve irrational numbers like square roots. Rationalizing the denominator is a technique used to eliminate radicals in the denominator to make fractions easier to handle and understand. In our example, \(\frac{3}{\sqrt{5}}\), multiplying by \(\sqrt{5}\) transforms the denominator into 5, a non-radical number. This results in the simpler fraction \(\frac{3\sqrt{5}}{5}\). Understanding the manipulation of fractions, therefore, is key to mastering algebraic expressions.

Algebra 2

Algebra 2 builds on basic algebraic concepts, including simplifying expressions, understanding functions, and working with complex numbers. A substantial part of Algebra 2 involves dealing with expressions that contain radicals and polynomials. Rationalizing denominators, as in \(\frac{3}{\sqrt{5}}\), is often encountered. This specific skill requires recognizing when and how to eliminate radicals to simplify expressions further. Mastering these techniques not only helps in solving equations but also deepens one’s understanding of the number system and mathematical logic. Algebra 2 serves as a foundation for higher mathematical studies, emphasizing problem-solving and analytical skills.