Problem 31 - A Graphical Approach to College Algebra

Question

If possible, simplify each radical expression. Assume that all variables represent positive real numbers. $$-\sqrt{\frac{9}{5}}$$

Step-by-Step Solution

Answer

The simplified form of \(-\sqrt{\frac{9}{5}}\) is \(-\frac{3\sqrt{5}}{5}\).

Step 1: Rewrite the Expression

The given expression is \(-\sqrt{\frac{9}{5}}\). We need to simplify this expression, which involves simplifying the radical part if possible.

Step 2: Simplify the Square Root

First, simplify the term under the square root. We want to look if the numerator and denominator can be simplified separately inside the square root. The square root of a fraction \(\frac{a}{b}\) is \(\frac{\sqrt{a}}{\sqrt{b}}\). Thus, \[\sqrt{\frac{9}{5}} = \frac{\sqrt{9}}{\sqrt{5}}.\]Next, simplify the square roots of the numerator and denominator separately.

Step 3: Calculate the Square Roots

Calculate \(\sqrt{9}\) and \(\sqrt{5}\). We know that \(\sqrt{9} = 3\), because 3 squared equals 9. The \(\sqrt{5}\) cannot be simplified further because 5 is a prime number. Hence, \[\sqrt{\frac{9}{5}} = \frac{3}{\sqrt{5}}.\]

Step 4: Rationalize the Denominator

We aim to have a simplified radical expression and typically, we try not to have radicals in the denominator. To eliminate \(\sqrt{5}\) from the denominator, multiply both the numerator and the denominator by \(\sqrt{5}\) to rationalize the fraction. Thus,\[\frac{3}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{3 \cdot \sqrt{5}}{5}.\]

Step 5: Finalize and Include the Negative Sign

Bring back the initial negative sign in the expression and write the entire fraction with the simplified terms: \( -\frac{3\sqrt{5}}{5} \). This is the simplified form of the original expression.

Key Concepts

Square Root Simplification

Square root simplification is the process of finding and expressing numbers under the square root in their simplest form. When we work with square roots, the goal is to break down a number into its simplest components, making calculations easier to handle.
To simplify a square root:

• Look for perfect squares in the number you are going to simplify. Perfect squares are numbers like 4, 9, 16, 25, etc., since these can be expressed as the square of whole numbers.
• Break the number under the square root into these perfect squares. For example, \( \sqrt{36} \) simplifies to \( 6 \) because 36 is \( 6^2 \).
• Express numbers with only non-perfect square components in the simplest radical form if possible.

By practicing square root simplification, you gain confidence in handling radical expressions.

Rationalizing the Denominator

Rationalizing the denominator is a process used in simplification where we eliminate radicals from the bottom of a fraction. The aim is to rewrite the fraction in conventional form where there are no square roots in the denominator. Here's how it can be done:

• Identify the radical in the denominator. If it's a simple \( \sqrt{n} \), multiply both the numerator and the denominator of the fraction by \( \sqrt{n} \).
• If the denominator is a binomial involving radicals, you might need to use conjugates. For example, if you have \( a + \sqrt{b} \), you multiply by its conjugate \( a - \sqrt{b} \).
• After multiplying, simplify the expression to ensure all radicals are cleared from the denominator.

This process makes the expression easier to interpret and use, particularly in more complex mathematical equations.

Fraction Simplification

Fraction simplification involves reducing fractions to their simplest form. It's a fundamental skill in mathematics, and it helps simplify calculations and avoid errors.
Here's a step-by-step guide:

• Identify the greatest common divisor (GCD) of the numerator and denominator.
• Divide both the numerator and the denominator by the GCD.
• Ensure the denominator is positive: sometimes simplification results in a negative denominator, flipping both signs can help it become positive.

Fraction simplification is straightforward but crucial. Practicing this regularly will enhance your ability to solve math problems quickly and accurately.