Problem 103 - Fundamentals of Mathematics

Question

For the following problems, find each value. Reduce answers to lowest terms or convert to mixed numbers. $$\frac{2}{3} \cdot \sqrt{\frac{9}{16}}$$

Step-by-Step Solution

Answer

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

Step 1: Simplify the Square Root

First, simplify the square root \( \sqrt{\frac{9}{16}} \). We know that \( \sqrt{\frac{9}{16}} = \frac{\sqrt{9}}{\sqrt{16}} \). Taking the square root of the numerator, \( \sqrt{9} = 3 \), and the square root of the denominator, \( \sqrt{16} = 4 \), we get \( \frac{3}{4} \).

Step 2: Multiply the Fractions

Now that we have simplified the square root to \( \frac{3}{4} \), multiply it by the other fraction: \( \frac{2}{3} \cdot \frac{3}{4} \). To multiply fractions, multiply the numerators together and the denominators together: \( \frac{2 \cdot 3}{3 \cdot 4} = \frac{6}{12} \).

Step 3: Simplify the Result

Next, simplify the fraction \( \frac{6}{12} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 6: \( \frac{6 \div 6}{12 \div 6} = \frac{1}{2} \).

Key Concepts

Simplifying Square Roots

Square roots can sometimes look intimidating, but simplifying them can be quite straightforward. When you have a square root of a fraction, like \( \sqrt{\frac{9}{16}} \), you can break it down into more manageable parts. This means finding the square root of the numerator and the square root of the denominator separately.

For example:

• The square root of the numerator \( \sqrt{9} \) is 3 because 3 times 3 equals 9.
• The square root of the denominator \( \sqrt{16} \) is 4 since 4 times 4 equals 16.

Now, after simplifying, \( \sqrt{\frac{9}{16}} \) becomes \( \frac{3}{4} \).

This process is a crucial step as it converts a complex-looking fraction into something much simpler. This allows us to easily work with the fraction in further calculations.

Multiplying Fractions

Multiplying fractions is simpler than you might think. Once the square root has been simplified to a fraction, you can handle multiplication easily. Consider multiplying \( \frac{2}{3} \) by \( \frac{3}{4} \). Follow these steps to make things clear:

• Multiply the numerators together: \( 2 \times 3 = 6 \).
• Multiply the denominators together: \( 3 \times 4 = 12 \).

So, the product of \( \frac{2}{3} \cdot \frac{3}{4} \) becomes \( \frac{6}{12} \).

Remember, when multiplying fractions, you don't need a common denominator as in addition or subtraction, which makes it quite straightforward. Multiplying across gives you the correct result effortlessly.

Simplifying Fractions

The final step to completing our problem involves simplifying the fraction obtained from multiplication. We begin with \( \frac{6}{12} \). To simplify, you need to find the greatest common divisor (GCD) for the numerator and the denominator.

In this case, the GCD of 6 and 12 is 6. Simplify the fraction by dividing the numerator and the denominator by the GCD:

• \( 6 \div 6 = 1 \)
• \( 12 \div 6 = 2 \)

Thus, \( \frac{6}{12} \) simplifies to \( \frac{1}{2} \).

Simplifying fractions is an essential skill to ensure the answer is in its simplest form, making it more understandable and often required in math assignments. With practice, this will become second nature and help make solving problems more efficient.