Problem 18 - Fundamentals of Mathematics

Question

Find each value. \((\) Section 4.5\() \frac{11}{30} \cdot \sqrt{\frac{225}{121}}\)

Step-by-Step Solution

Answer

The value is \( \frac{1}{2} \).

Step 1: Simplify the Square Root Expression

The expression inside the square root is \( \frac{225}{121} \). The square root of a fraction \( \frac{a}{b} \) is equal to \( \frac{\sqrt{a}}{\sqrt{b}} \). Therefore, we have \( \sqrt{\frac{225}{121}} = \frac{\sqrt{225}}{\sqrt{121}} \).

Step 2: Calculate the Square Roots

Find the square roots of the numerator and the denominator. \( \sqrt{225} = 15 \) and \( \sqrt{121} = 11 \). Substitute these values back into the fraction to get \( \frac{15}{11} \).

Step 3: Multiply the Expressions

Now substitute the simplified square root expression back into the original equation: \( \frac{11}{30} \cdot \frac{15}{11} \). Multiply the numerators and the denominators: \( \frac{11 \cdot 15}{30 \cdot 11} = \frac{165}{330} \).

Step 4: Simplify the Fraction

The fraction \( \frac{165}{330} \) can be simplified by finding the greatest common divisor (GCD) of 165 and 330. The GCD is 165. Divide both numerator and denominator by 165 to get \( \frac{1}{2} \).

Key Concepts

Simplifying Fractions

Simplifying fractions is an important skill in mathematics because it helps you express fractions in their simplest form. A fraction is simplified when the numerator and the denominator have no common factors other than 1. This is achieved by dividing both the numerator and the denominator by their greatest common divisor (GCD).

To simplify a fraction:

• Identify the greatest common divisor of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.
• The result is a fraction in its simplest form.

For example, in the step-by-step solution where the fraction \( \frac{165}{330} \) was simplified, the GCD of 165 and 330 was found to be 165. Dividing both by this number simplified the fraction to \( \frac{1}{2} \).

Simplifying fractions efficiently makes calculations easier and highlights the core relationships between numbers.

Multiplying Fractions

Multiplying fractions involves multiplying the numerators together to get a new numerator and the denominators together to get a new denominator. This is an essential technique used in many areas of math.

To multiply two fractions:

• Multiply the numerators of the fractions to get the new numerator.
• Multiply the denominators to get the new denominator.
• Simplify the resulting fraction, if possible.

In the problem provided, multiplying \( \frac{11}{30} \) by \( \frac{15}{11} \) meant multiplying 11 by 15 and 30 by 11, resulting in \( \frac{165}{330} \).

The multiplication process is straightforward, but it's always a good idea to simplify the result when possible. This not only provides a neater answer but also improves understanding of the fraction's value.

Square Roots

Square roots are used to find a number that, when multiplied by itself, gives the original number under the square root sign. Working with square roots of fractions involves taking the square root of the numerator and the denominator separately.

Here's how to manage square roots in fractions effectively:

• The square root of a fraction \( \frac{a}{b} \) is interpreted as \( \frac{\sqrt{a}}{\sqrt{b}} \).
• Calculate the square root of both the numerator and the denominator.
• Simplify the resulting fraction if needed.

In this exercise, the expression \( \sqrt{\frac{225}{121}} \) was simplified to \( \frac{\sqrt{225}}{\sqrt{121}} \), which is \( \frac{15}{11} \).

Recognizing and utilizing the properties of square roots can simplify expressions and yield more manageable calculations.