Problem 41 - Prealgebra

Question

Compute the exact square root. \(\sqrt{\frac{121}{49}}\)

Step-by-Step Solution

Answer

The exact square root is \( \frac{11}{7} \).

Step 1: Recognize the Expression Inside the Square Root

The given expression is \( \sqrt{\frac{121}{49}} \), which is the square root of a fraction. Our task is to find the exact value of this expression.

Step 2: Apply the Property of Square Roots on Fractions

We know that \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \). So we can separate the square root of the numerator and denominator. Let's apply this property: \[ \sqrt{\frac{121}{49}} = \frac{\sqrt{121}}{\sqrt{49}} \]

Step 3: Determine the Exact Square Roots of the Numerator and Denominator

Now, we calculate the square root of 121 and the square root of 49 separately. We recognize that 121 is \( 11^2 \), and 49 is \( 7^2 \). Therefore: \[ \sqrt{121} = 11 \] and \[ \sqrt{49} = 7 \].

Step 4: Compute the Exact Value of the Expression

Substitute the square roots calculated in the previous step back into the fraction: \[ \frac{\sqrt{121}}{\sqrt{49}} = \frac{11}{7} \]. Thus, the exact value of \( \sqrt{\frac{121}{49}} \) is \( \frac{11}{7} \).

Key Concepts

Fraction Simplification

Fraction simplification involves reducing a fraction to its simplest form. A fraction consists of a numerator and a denominator. In its simplest form, both the numerator and the denominator should have no common factors other than 1.
This is achieved by dividing both the numerator and the denominator by their greatest common divisor (GCD).
In the example of simplifying the fraction \( \frac{11}{7} \):

• The GCD of 11 and 7 is 1, meaning there are no whole numbers greater than 1 that divide both of them evenly.
• Thus, the fraction \( \frac{11}{7} \) is already in its simplest form.

Reducing fractions is useful because it makes them easier to understand and work with in mathematical expressions.

Numerator and Denominator

Understanding the roles of the numerator and denominator is fundamental when working with fractions. The numerator is the number above the fraction bar, while the denominator is the number below it.
The numerator represents how many parts of the whole are being considered, whereas the denominator shows the total number of equal parts that make up the whole.
In the fraction \( \frac{11}{7} \):

• 11 is the numerator, indicating 11 parts of something.
• 7 is the denominator, suggesting that there are 7 equal parts in total.

When taking the square root of a fraction, both the numerator and the denominator are examined separately to determine their respective square roots.

Properties of Square Roots

Square roots have unique properties that make them very useful in mathematics. One key property is that the square root of a fraction can be expressed as a fraction of square roots.
Specifically, the rule \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \) allows us to find the square root of each part of the fraction separately.
Let's consider the fraction \( \frac{121}{49} \):

• Separate the fraction into \( \sqrt{121} \) and \( \sqrt{49} \).
• Determine the square roots: \( \sqrt{121} = 11 \) and \( \sqrt{49} = 7 \).
• Put them back together to get \( \frac{11}{7} \).

This method is powerful for simplifying expressions where square roots are involved, leading directly to exact and simplified results.