Problem 13 - Elementary Algebra

Question

Simplify each square root. $$ \sqrt{\frac{9}{4}} $$

Step-by-Step Solution

Answer

Answer: The simplified square root is: $$\frac{3}{2}$$

Step 1: Simplify the fraction inside the square root

First, we need to simplify the fraction inside the square root. To do this, look for common factors in both the numerator and denominator. The fraction given is: $$ \frac{9}{4} $$ There are no common factors between 9 and 4, so this fraction is already in its simplest form.

Step 2: Find the square root of the simplified fraction

Now that the fraction is simplified, we can find the square root of the numerator and the square root of the denominator separately: $$ \sqrt{\frac{9}{4}} = \frac{\sqrt{9}}{\sqrt{4}} $$ We know that the square root of 9 is 3 (since 3 * 3 = 9) and the square root of 4 is 2 (since 2 * 2 = 4). So we get: $$ \frac{\sqrt{9}}{\sqrt{4}} = \frac{3}{2} $$ So the simplified square root is: $$ \sqrt{\frac{9}{4}} = \frac{3}{2} $$

Key Concepts

Square Roots

The square root symbol \( \sqrt{} \) represents a value that, when multiplied by itself, equals the original number under the square root sign. It's important to understand that square roots reverse the operation of squaring a number. For example, since 3 squared \((3^2)\) equals 9, the square root of 9 is 3.

When dealing with square roots of fractions, the principles remain the same. We apply the square root operation separately to both the numerator and the denominator of the fraction under the square root.

• If \( \sqrt{\frac{a}{b}} \), then it translates to \( \frac{\sqrt{a}}{\sqrt{b}} \).
• This property allows us to simplify complex square roots by simplifying the fraction first.

To ease calculation, always consider whether the numbers inside the square root are perfect squares or factorable, which can simplify the process.

Fractions

Fractions represent a part of a whole and are composed of two parts: the numerator and the denominator. Understanding their structure is crucial when performing operations such as finding square roots.

In any given fraction \( \frac{a}{b} \), the top number, \(a\), is the numerator, and the bottom number, \(b\), is the denominator. Fractions can be simplified by dividing both the numerator and the denominator by their greatest common factor (GCF).

• For example, in the fraction \( \frac{9}{4} \), 9 is the numerator, and 4 is the denominator.
• If 9 and 4 had common factors, they would be simplified by dividing both by the common factor.

While fractions might seem intimidating at first, they become much easier to work with once simplified.

Numerator and Denominator

The numerator and denominator are fundamental parts of a fraction that determine its value. They are essential when applying operations like division, multiplication, or finding square roots to fractions.

The numerator tells us how many parts of a whole are being considered, while the denominator shows how many equal parts the whole is divided into. Simplifying these can vastly change the manipulability of the fraction, especially when working with square roots.

• It is often necessary to address the numerator and the denominator individually, especially in operations involving square roots.
• In \( \frac{9}{4} \), we determine the square root of 9 as \(3\) and the square root of 4 as \(2\), hence \( \frac{3}{2} \).

Understanding both elements not only aids in simplification but also enhances overall mathematical proficiency in handling fractions.