Problem 76 - Algebra 2

Question

Simplify each expression. \(\frac{4}{\sqrt{6}}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{2\sqrt{6}}{3}\).

Step 1: Identify the Expression

We start with the expression \(\frac{4}{\sqrt{6}}\). Our goal is to simplify this expression by removing the square root from the denominator.

Step 2: Rationalize the Denominator

To eliminate the square root from the denominator, we multiply both the numerator and the denominator by the square root of 6. This step is important because it rationalizes the denominator. Our expression becomes: \(\frac{4}{\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}} = \frac{4\sqrt{6}}{6}\).

Step 3: Simplify the Expression

Now that we have rationalized the denominator, we look for any further simplification. We notice that the fraction \(\frac{4\sqrt{6}}{6}\) can be reduced by simplifying the coefficients. \(\frac{4}{6}\) simplifies to \(\frac{2}{3}\), so the expression becomes \(\frac{2\sqrt{6}}{3}\).

Key Concepts

Simplifying Expressions

When you simplify an expression, the aim is to make it easier to work with or understand. This is done by performing operations that reduce the expression to its simplest form without changing its value. One common way is to combine like terms or operate on fractions. Simplifying expressions helps in making further calculations easier and can reveal important characteristics of the problem at hand.

In our problem,

• we simplified \(rac{4}{rac{6}{2}}\) by first rationalizing the denominator and then reducing the fraction to \(rac{2rac{6}{2}}{3}\).

Thus, the process not only removed the square root from the denominator but also reduced the fraction for an easier interpretation.

Square Roots

Square roots represent one of the two equal factors of a number. For instance, \(rac{6}{2}\) is a square root of 36 because \(rac{6}{2} \times \frac{6}{2} = 36\). They often appear in mathematical expressions, especially when dealing with quadratic equations or when calculating areas.

Understanding how to manipulate square roots is vital because they can complicate expressions, especially when they appear in denominators. Such is the case in our example, \(rac{4}{rac{6}{2}}\), where the square root in the denominator requires us to rationalize it to simplify the expression. By multiplying by \(rac{6}{2} \over \frac{6}{2}\), the problematic square root is eliminated.

Fraction Simplification

Simplifying fractions means reducing them to their simplest form, which involves dividing the numerator and the denominator by their greatest common divisor. Simplification makes the fraction easier to understand and use in calculations.

In the expression we're dealing with, \(rac{4\sqrt{6}}{6}\) can be simplified:

• The fraction \(rac{4}{6}\) was reduced to \(rac{2}{3}\)
• The simplified expression becomes \(rac{2\sqrt{6}}{3}\)

Fraction simplification is an essential skill in mathematics as it often makes solutions easier and clearer, particularly in problems involving multiple operations.