Problem 25 - College Algebra

Question

Use the quotient rule to simplify the expressions in Exercises. Assume that \(x>0.\) $$\sqrt{\frac{49}{16}}$$

Step-by-Step Solution

Answer

\(\frac{7}{4}\)

Step 1: Simplify the square root

The first step is to apply the square root to both the numerator and the denominator separately. As a result, we get \(\sqrt{\frac{49}{16}} = \frac{\sqrt{49}}{\sqrt{16}}\).

Step 2: Calculate the square root of both numerator and denominator

The square root of 49 is 7 (since \(7 \times 7 = 49\)) and the square root of 16 is 4 (since \(4 \times 4 = 16\)). Therefore, we get \(\frac{\sqrt{49}}{\sqrt{16}} = \frac{7}{4}\).

Key Concepts

Quotient Rule for Radicals

Understanding the quotient rule for radicals is essential when simplifying expressions that involve square roots. This rule states that the square root of a quotient is equal to the quotient of the square roots of the numerator and denominator. In mathematical terms,
\[ \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \]
for any nonnegative numbers a and b, where b is not zero since division by zero is undefined.

Application of the Quotient Rule

To apply this rule, you divide the problem into two simpler square root problems, one for the numerator and one for the denominator. This concept is practically used when we have a fraction under a radical sign, and we want to simplify it. It is important to note that the rule can only be applied if the original square roots are defined; in other words, we cannot have a negative number underneath the square root when dealing with real numbers as that would result in an imaginary number.

Understanding Radicals

Radicals are mathematical expressions that include a root, the most common of which is the square root. The square root of a number is a value that, when multiplied by itself, gives the original number. Symbolically, it is represented as \( \sqrt{x} \).
Radicals can involve variables as well as numbers, and they follow specific rules for simplification and manipulation. A radical expression might look complicated, but understanding the properties of exponents and roots helps to handle them with ease.

Properties of Radicals

Some important properties include the product rule for radicals \( \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \)and the aforementioned quotient rule. When dealing with radicals, ensure the number or expression inside the radical is as simplified as possible. This could mean removing factors that are perfect squares or simplifying any rational numbers inside the radical.

Square Root Simplification

Simplifying square roots, or finding their most simplified form, involves identifying and extracting perfect squares from the radicand—the number inside the square root. The basic idea is to make the radical as simple as possible.
For example, \( \sqrt{48} \)can be simplified by recognizing that 48 is divisible by the perfect square 16. The expression becomes \( \sqrt{16 \cdot 3} \), which simplifies further to \( 4\sqrt{3} \).

Finding Simplified Square Roots

When simplifying square roots:

• Look for the largest perfect square factor of the radicand.
• Express the radicand as the product of this factor and another number.
• Apply the square root separately to the perfect square and the remaining factor.
• Simplify the expression by multiplying the square root of the perfect square by the square root of the remaining factor.

In our problem, the radicands 49 and 16 are already perfect squares, which makes simplification straightforward because their square roots are whole numbers.