Problem 74 - Algebra 2

Question

Simplify each expression. \(\frac{2}{\sqrt{3}}\)

Step-by-Step Solution

Answer

\(\frac{2\sqrt{3}}{3}\)

Step 1: Identify the Problem

We have the expression \( \frac{2}{\sqrt{3}} \). The goal is to simplify it by rationalizing the denominator.

Step 2: Rationalizing the Denominator

To eliminate the square root in the denominator, we multiply both the numerator and the denominator by \( \sqrt{3} \). This gives us \( \frac{2 \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}} \).

Step 3: Simplify the Denominator

\( \sqrt{3} \times \sqrt{3} = 3 \), so the denominator becomes 3. Now the expression is \( \frac{2\sqrt{3}}{3} \).

Step 4: Write the Simplified Form

The simplified form of the expression is \( \frac{2\sqrt{3}}{3} \). This is in its simplest rationalized form without any square roots in the denominator.

Key Concepts

Simplifying Expressions

Simplifying expressions is a key skill in algebra that involves rewriting complex expressions in a simpler or more convenient form while maintaining their original value. It's like cleaning up your room to make it neat and organized. In mathematics, simplifying can make it easier to handle equations and solve problems.When simplifying, we aim to:

• Reduce the expression to its simplest terms.
• Remove any unnecessary complexity, such as square roots in the denominator.
• Make calculations more straightforward and avoid potential errors during operations.

It is essential to follow a systematic approach like simplifying each part of the expression step-by-step and ensuring the value of the expression remains unchanged.For example, in simplifying the expression \(\frac{2}{\sqrt{3}}\), multiplying by \(\sqrt{3}\) was a strategic choice to eliminate the square root in the denominator, leading us to a cleaner and more manageable form \(\frac{2\sqrt{3}}{3}\).

Square Roots

Square roots are quite a fascinating topic in mathematics. A square root of a number is a value that, when multiplied by itself, gives the original number. It’s symbolized by the radical sign \(\sqrt{\cdot}\). Understanding square roots is crucial for simplifying expressions, especially when they appear in the denominator of a fraction.A useful property of square roots is that \( \sqrt{a} \times \sqrt{a} = a \). This principle is particularly helpful when rationalizing denominators, as it allows us to eliminate square roots by creating entire numbers. In the expression \(\frac{2}{\sqrt{3}}\), notice how multiplying the numerator and the denominator by \(\sqrt{3}\) effectively uses this property, transforming \(\sqrt{3} \times \sqrt{3}\) into a neat \(3\). This manipulation is why we can convert expressions into simpler forms without introducing errors.

Fraction Operations

Working with fractions is an essential part of math. Fraction operations include tasks like addition, subtraction, multiplication, and division of fractions. A critical aspect of fraction operations is making sure fractions are in their simplest form, which includes rationalizing denominators.Rationalizing denominators, like in \(\frac{2}{\sqrt{3}}\), involves removing irrational numbers such as square roots from the denominator. This is done by multiplying both the numerator and the denominator by a number that will cancel out the square root in the denominator.Consider the following tips when handling fraction operations:

• Always seek to keep fractions simplified.
• Multipy both parts by the same value to maintain the fraction's equality.
• Stay aware of properties of numbers like square roots and whole numbers.

With this knowledge, handling and simplifying even complex-looking fractions becomes an easy task.