Problem 19 - 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. $$ 14 \sqrt{\frac{1}{7}}+\sqrt{28} $$

Step-by-Step Solution

Answer

The expression simplifies to \(4\sqrt{7}\).

Step 1: Simplify Inside the Square Roots

Start with the expression \(\sqrt{\frac{1}{7}}\). We know the square root of a fraction is the square root of the numerator over the square root of the denominator. Therefore, \(\sqrt{\frac{1}{7}} = \frac{\sqrt{1}}{\sqrt{7}} = \frac{1}{\sqrt{7}}\).

Step 2: Rationalize the First Term

Multiply both the numerator and the denominator of \(\frac{1}{\sqrt{7}}\) by \(\sqrt{7}\) to eliminate the square root in the denominator: \(\frac{1}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}} = \frac{\sqrt{7}}{7}\). Thus, \(14 \sqrt{\frac{1}{7}} = 14 \times \frac{\sqrt{7}}{7} = 2 \sqrt{7}\).

Step 3: Simplify the Second Term

For \(\sqrt{28}\), rewrite 28 as a product of integers: \(\sqrt{28} = \sqrt{4 \times 7} = \sqrt{4} \times \sqrt{7}\). Since \(\sqrt{4} = 2\), it can be simplified to \(2\sqrt{7}\).

Step 4: Combine Like Terms

Now that both terms are in terms of \(\sqrt{7}\), combine them: \(2\sqrt{7} + 2\sqrt{7} = 4\sqrt{7}\). Thus, the expression simplifies to \(4\sqrt{7}\).

Key Concepts

Rationalizing the Denominator

When dealing with fractions, especially those involving square roots, it's often useful to "rationalize the denominator." This process involves eliminating square roots from the denominator.

In the given example, we started with the fraction \( \frac{1}{\sqrt{7}} \). Noticing the square root in the denominator, we multiplied both the numerator and the denominator by \( \sqrt{7} \).

This is a clever trick because multiplying \( \sqrt{7} \cdot \sqrt{7} \) gives just 7, a whole number. This modification simplifies the expression to \( \frac{\sqrt{7}}{7} \). Thus, the denominator no longer houses a square root but a simple whole number, making the expression neater and often easier to work with in both algebra and further calculations.

Combining Like Terms

Once all individual square root expressions are simplified and rationalized, the next step is to combine them if they are 'like terms'.

In this context, like terms are expressions that have the same square root component. In our case, the simplified terms \( 2\sqrt{7} \) and another \( 2\sqrt{7} \) are like terms. They both share \( \sqrt{7} \) as the radical, making them compatible for addition.

Combining them is straightforward:

• Simply add the coefficients of the like terms.
• \( 2\sqrt{7} \) plus \( 2\sqrt{7} \) equals \( (2 + 2)\sqrt{7} = 4\sqrt{7} \).

By combining the terms, you achieve a single, simplified expression, which is cleaner and often easier to interpret or use in further operations.

Square Roots of Fractions

Dealing with square roots of fractions involves breaking down the fraction into more manageable parts. The square root of a fraction like \( \sqrt{\frac{1}{7}} \) is simplified by applying the square root to both the numerator and the denominator separately.

This means that \( \sqrt{\frac{1}{7}} \) can be expressed as \( \frac{\sqrt{1}}{\sqrt{7}} \). Since \( \sqrt{1} \) is simply 1, the fraction simplifies to \( \frac{1}{\sqrt{7}} \).

This operation is a basic property of square roots that states \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \). Understanding this rule enables you to systematically approach and simplify any square root of a fraction, which is a common challenge in algebraic manipulations.