Problem 102 - Fundamentals of Mathematics

Question

For the following problems, find each value. Reduce answers to lowest terms or convert to mixed numbers. $$\sqrt{\frac{144}{25}}$$

Step-by-Step Solution

Answer

\( \sqrt{\frac{144}{25}} = 2 \frac{2}{5} \).

Step 1: Simplify the Fraction Inside the Square Root

The given expression is \( \sqrt{\frac{144}{25}} \). You can treat the square root of a fraction as the square root of the numerator divided by the square root of the denominator. Hence, this becomes \( \frac{\sqrt{144}}{\sqrt{25}} \).

Step 2: Calculate the Square Roots of Numerator and Denominator

Now, calculate the square root of the numerator, \( \sqrt{144} \), which is 12, since \( 12 \times 12 = 144 \). Then, calculate the square root of the denominator, \( \sqrt{25} \), which is 5 because \( 5 \times 5 = 25 \). Hence, the expression becomes \( \frac{12}{5} \).

Step 3: Express as a Mixed Number

The fraction \( \frac{12}{5} \) can be expressed as a mixed number. Divide 12 by 5, which gives you 2 with a remainder of 2. Therefore, \( \frac{12}{5} = 2 \frac{2}{5} \).

Key Concepts

Square Root of a Fraction

Understanding how to find the square root of a fraction is a handy skill. A fraction is composed of two parts: the numerator (top number) and the denominator (bottom number). To find the square root of a fraction like \( \sqrt{\frac{144}{25}} \), you can treat it much like regular numbers, only that you handle each part separately.

Here's the straightforward method to do it:

• First, take the square root of the numerator. In this example, the numerator is 144. The square root of 144 is 12, since \( 12 \times 12 = 144 \).
• Next, calculate the square root of the denominator. Our denominator here is 25, and its square root is 5 because \( 5 \times 5 = 25 \).
• Put them together into a fraction, so \( \sqrt{\frac{144}{25}} = \frac{12}{5} \).

This method allows you to break down what could be complex into simple steps, making calculating easier and more intuitive. Once you have the fraction, you may need to further simplify or convert it, depending on what the problem asks.

Converting Improper Fractions to Mixed Numbers

Sometimes, you'll end up with an improper fraction like \( \frac{12}{5} \). An improper fraction means that the numerator is larger than the denominator. To make these easier to understand or visualize, you can convert them into mixed numbers.

Here's how that works:

• Divide the numerator by the denominator. For \( \frac{12}{5} \), divide 12 by 5. The result is 2 with a remainder.
• The whole number you get from this division (in this case, 2) becomes the whole part of your mixed number.
• The remainder becomes the new numerator, and the original denominator stays the same, giving you \( 2 \frac{2}{5} \).

Conversions like this help make the number more tangible, especially when you are dealing with measurements or other practical situations.

Lowest Terms

Simplifying fractions to their lowest terms is an important mathematical process. A fraction is in its simplest form when the numerator and denominator have no common divisors other than 1. This makes it easier to understand and utilize.

To simplify a fraction like \( \frac{12}{5} \), which is already in its lowest terms considering the numbers involved:

• Examine both numbers to ensure they have no other common factors. In this example, 12 and 5 share no divisors except 1.
• If there were common factors, you would divide both numbers by their greatest common divisor (GCD).

While our example of \( \frac{12}{5} \) doesn’t require any simplification, knowing how to reduce fractions is crucial, especially when dealing with complex equations or trying to find a neat answer.