Problem 26 - Algebra 2 and Trigonometry

Question

In \(11-38,\) evaluate each expression in the set of real numbers. $$ -\sqrt{\frac{49}{36}} $$

Step-by-Step Solution

Answer

The value of the expression is \(-\frac{7}{6}\).

Step 1: Understand the Expression

The given expression is \[-\sqrt{\frac{49}{36}}\]which asks us to find the square root of the fraction \(\frac{49}{36}\) and then apply a negative sign to the result.

Step 2: Simplify the Fraction Inside the Square Root

Recognize that \(\frac{49}{36}\) is a fraction. The number 49 is a perfect square, as is 36. This fraction can be expressed as \(\left(\frac{7}{6}\right)^2\), meaning that it is the square of \(\frac{7}{6}\).

Step 3: Take the Square Root

Since \(\frac{49}{36}\) is \(\left(\frac{7}{6}\right)^2\), the square root of \(\frac{49}{36}\) is simply \(\frac{7}{6}\). Hence, \[\sqrt{\frac{49}{36}} = \frac{7}{6}.\]

Step 4: Apply the Negative Sign

Use the negative sign from the original expression. The expression then becomes:\[-\frac{7}{6}.\]

Step 5: Final Answer

Combine these steps to conclude that the evaluated expression is:\[-\frac{7}{6}.\]

Key Concepts

Simplifying Fractions

Fractions may appear intimidating at first, but simplifying them is straightforward and crucial for solving mathematical problems. A fraction consists of two parts: a numerator (the number on top) and a denominator (the number on bottom). The goal in simplifying fractions is to express the fraction in its simplest form, making it easier to handle in calculations.
One way to simplify a fraction is by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD is the largest number that divides both numbers without leaving a remainder. For example, to simplify the fraction \( \frac{49}{36} \), we begin by determining if the numerator and denominator share any common factors. In this case, both 49 and 36 are perfect squares, meaning 49 is \(7^2\) and 36 is \(6^2\).
By recognizing that these numbers can be expressed as squares, we can rewrite \( \frac{49}{36} \) as \( \left( \frac{7}{6} \right)^2 \). This form revelation is not just a simplification of the individual numbers, but it also significantly assists in the next steps concerning operations like finding square roots.

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. In notation, the square root of a number \( x \) is represented as \( \sqrt{x} \). For instance, \( \sqrt{49} = 7 \) and \( \sqrt{36} = 6 \). With fractions, the square root operation is slightly different but follows similar logic.
When you take the square root of a fraction, you take the square root of the numerator and the denominator separately. For the fraction \( \frac{49}{36} \), since we know it can be expressed as \( \left( \frac{7}{6} \right)^2 \), taking the square root becomes straightforward. Essentially, you "remove" the square:

• The square root of \( 49 \) (the numerator) is \( 7 \).
• The square root of \( 36 \) (the denominator) is \( 6 \).

Thus, \( \sqrt{\frac{49}{36}} = \frac{7}{6} \). Discovering this relationship leverages the understanding that both 49 and 36 are perfect squares, simplifying the computation greatly.

Evaluating Expressions

Evaluating expressions involves solving or simplifying a mathematical expression to a simpler form. With the expression \( -\sqrt{\frac{49}{36}} \), the task is to systematically break down and resolve the given components using mathematical operations.
We started by simplifying the fraction under the square root, recognizing that \( \frac{49}{36} = \left( \frac{7}{6} \right)^2 \). From there, we took the square root, knowing that \( \sqrt{\left( \frac{7}{6} \right)^2} = \frac{7}{6} \). The final step in evaluating the expression is applying any operations outside the initial square root operation. In this case, it is the negative sign.
The negative sign simply instructs us to take the positive result of \( \frac{7}{6} \) and multiply it by \(-1\), resulting in \(-\frac{7}{6} \). Thus, to evaluate such expressions, follow this logical structure:

• Simplify inside the radical or parentheses first where possible.
• Take the square root if applicable.
• Apply any other operations, such as negative signs, to finalize the result.

This structured approach helps manage each part of the expression carefully, ensuring a correct outcome.