Problem 17 - Fundamentals of Mathematics

Question

Find each value. \((\) Section 4.5\() \sqrt{\frac{4}{9}} \cdot \sqrt{\frac{81}{64}}\)

Step-by-Step Solution

Answer

The value is \( \frac{3}{4} \).

Step 1: Simplify Each Square Root

First, we'll simplify each square root separately. We have two expressions: 1. \( \sqrt{\frac{4}{9}} \)2. \( \sqrt{\frac{81}{64}} \)We need to find the square root of each fraction.

Step 2: Sub-step 1.1: Simplify \( \sqrt{\frac{4}{9}} \)

\( \sqrt{\frac{4}{9}} = \frac{\sqrt{4}}{\sqrt{9}} = \frac{2}{3} \). The square root of 4 is 2, and the square root of 9 is 3, so \( \sqrt{\frac{4}{9}} \) simplifies to \( \frac{2}{3} \).

Step 3: Sub-step 1.2: Simplify \( \sqrt{\frac{81}{64}} \)

\( \sqrt{\frac{81}{64}} = \frac{\sqrt{81}}{\sqrt{64}} = \frac{9}{8} \). The square root of 81 is 9, and the square root of 64 is 8, so \( \sqrt{\frac{81}{64}} \) simplifies to \( \frac{9}{8} \).

Step 2: Multiply the Simplified Fractions

Now multiply the simplified expressions from Sub-steps 1.1 and 1.2:\[ \frac{2}{3} \times \frac{9}{8} \]Multiply the numerators and then the denominators:\[ \frac{2 \times 9}{3 \times 8} = \frac{18}{24} \]

Step 3: Simplify the Resulting Fraction

Finally, simplify the fraction \( \frac{18}{24} \). Both the numerator and the denominator can be divided by 6:\[ \frac{18 \div 6}{24 \div 6} = \frac{3}{4} \]. So the simplified result is \( \frac{3}{4} \).

Key Concepts

Square Roots of Fractions

When dealing with square roots of fractions, the process is simpler than it might first appear. You can understand it by breaking down the square root of a fraction into two separate parts: the numerator and the denominator. This is due to an important property of square roots:

• If you have the fraction \( \frac{a}{b} \), the square root of this fraction can be represented as \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \).

Thus, for \( \sqrt{\frac{4}{9}} \), it becomes \( \frac{\sqrt{4}}{\sqrt{9}} \), which simplifies to \( \frac{2}{3} \). This is because \( \sqrt{4} = 2 \) and \( \sqrt{9} = 3 \). Similarly, for \( \sqrt{\frac{81}{64}} \), you break it down into \( \frac{\sqrt{81}}{\sqrt{64}} \), which simplifies to \( \frac{9}{8} \). Understanding this method allows you to manage square roots of fractions effectively, breaking them down into simpler, workable parts.

Fraction Multiplication

Once you have simplified the square roots into fractions, the next task is to multiply them. Fraction multiplication is straightforward. Here’s how it works:

• Multiply the numerators (top numbers) together.
• Multiply the denominators (bottom numbers) together.

For example, if we take the simplified results from our square roots, we multiply \( \frac{2}{3} \) by \( \frac{9}{8} \). This results in:

• Numerator: \( 2 \times 9 = 18 \)
• Denominator: \( 3 \times 8 = 24 \)

Thus, before simplification, \( \frac{2}{3} \times \frac{9}{8} = \frac{18}{24} \). Each step is simple calculation, but essential to getting the right result. Remember, the order of the multiplication doesn’t matter for fractions, which makes these operations easy to handle.

Simplifying Fractions

The final step in solving problems involving fraction multiplication is simplifying the resulting fraction. Simplification involves reducing the fraction to its simplest form where the numerator and denominator have no common divisors other than 1. To simplify \( \frac{18}{24} \), look for the greatest common divisor (GCD) of the numerator and the denominator.

• The GCD of 18 and 24 is 6.

You divide both the numerator and the denominator by their GCD:

• \( \frac{18}{24} \rightarrow \frac{18 \div 6}{24 \div 6} = \frac{3}{4} \)

Thus, \( \frac{3}{4} \) is the simplest form of \( \frac{18}{24} \). Simplifying fractions makes them easier to understand and compare, keeping calculations neat and manageable. This process is vital for working efficiently with fractions.