Problem 25 - Fundamentals of Mathematics

Question

Find the value of each of the following. $$ \frac{3}{8} \cdot \sqrt{\frac{1}{9}} $$

Step-by-Step Solution

Answer

The value is \( \frac{1}{8} \).

Step 1: Simplify the Square Root

First, evaluate \( \sqrt{\frac{1}{9}} \). The square root of a fraction is the square root of the numerator divided by the square root of the denominator. So, \( \sqrt{\frac{1}{9}} = \frac{\sqrt{1}}{\sqrt{9}} = \frac{1}{3} \).

Step 2: Multiply the Fractions

Now, multiply the fractions \( \frac{3}{8} \) and \( \frac{1}{3} \). This involves multiplying the numerators and the denominators separately: \( \frac{3 \times 1}{8 \times 3} = \frac{3}{24} \).

Step 3: Simplify the Product

Finally, simplify \( \frac{3}{24} \) by finding the greatest common divisor of 3 and 24, which is 3. Divide both the numerator and the denominator by 3: \( \frac{3 \div 3}{24 \div 3} = \frac{1}{8} \).

Key Concepts

Square Root of a Fraction

To find the square root of a fraction, you simply take the square root of both the numerator and the denominator separately. This can be very useful, especially when dealing with fractions where both the numerator and the denominator are perfect squares.
For instance, let's consider the fraction \( \frac{1}{9} \). It's important to recognize that both numbers involved are perfect squares. The numerator is 1, and the denominator is 9.

• Find \( \sqrt{1} \) which is 1, since 1 is a perfect square.
• Find \( \sqrt{9} \) which is 3, because 3 times 3 equals 9.
• Therefore, \( \sqrt{\frac{1}{9}} = \frac{1}{3} \).

Understanding this step lays a foundation for many algebraic calculations and can make more complex problems simpler by isolating smaller, more manageable parts.

Simplifying Fractions

Simplifying fractions allows us to express a fraction in its simplest form. This process involves dividing the numerator and the denominator by their greatest common divisor (GCD).
Why is this important? It helps in making calculations easier and the results more understandable. Let's look at the straightforward example of simplifying \( \frac{3}{24} \).

• First, recognize that both 3 and 24 are divisible by the same number, which is a step in finding the GCD.
• Since 3 is a factor of 24, we use this knowledge to simplify.
• Divide 3 by 3 to get 1, and divide 24 by 3 to get 8.
• Thus, \( \frac{3}{24} \) simplifies to \( \frac{1}{8} \).

Simplifying isn't always obvious at first, but with practice, it becomes a quick, second-nature step in solving fraction problems.

Greatest Common Divisor

When simplifying fractions, knowing how to find the greatest common divisor (GCD) is incredibly handy. The GCD is the largest number that divides both the numerator and the denominator of a fraction without a remainder.
To better understand how this works, let's take the example of the fraction \( \frac{3}{24} \).

• List the factors of 3: 1, 3.
• List the factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
• The largest number common to both lists is 3, thus the GCD is 3.
• Divide both the numerator and the denominator by this common divisor to simplify the fraction.

Finding the GCD allows us to reduce fractions to their simplest terms, making them easier to understand and use in calculations. It's a basic yet crucial tool in mathematics.