Problem 104 - Fundamentals of Mathematics

Question

For the following problems, find each value. Reduce answers to lowest terms or convert to mixed numbers. $$\frac{3}{5} \cdot \sqrt{\frac{25}{81}}$$

Step-by-Step Solution

Answer

\( \frac{1}{3} \)

Step 1: Simplify the Square Root

First, simplify the square root in the expression. We have \( \sqrt{\frac{25}{81}} \). To find this, take the square root of the numerator and the denominator separately. \( \sqrt{25} = 5 \) and \( \sqrt{81} = 9 \). Therefore, \( \sqrt{\frac{25}{81}} = \frac{5}{9} \).

Step 2: Multiply the Fractions

Now, multiply \( \frac{3}{5} \) by the simplified result of the square root, which is \( \frac{5}{9} \). Use the formula for multiplying fractions: \( \frac{a}{b} \times \frac{c}{d} = \frac{a \cdot c}{b \cdot d} \). Here, \( a = 3, b = 5, c = 5, d = 9 \). So, \( \frac{3 \times 5}{5 \times 9} = \frac{15}{45} \).

Step 3: Reduce the Fraction to Lowest Terms

Finally, reduce \( \frac{15}{45} \) to its lowest terms. Both the numerator and the denominator can be divided by their greatest common divisor, which is 15. \( \frac{15 \div 15}{45 \div 15} = \frac{1}{3} \).

Key Concepts

Simplifying Square Roots

Simplifying square roots is all about breaking down the components under the square root so they are more manageable. In mathematics, square roots show us a number which, when multiplied by itself, gives the original number.
To simplify a fraction under a square root, like \( \sqrt{\frac{25}{81}} \), take the square root of both the numerator and the denominator separately:

• Find the square root of 25, which is 5, because \( 5 \times 5 = 25 \).
• Find the square root of 81, which is 9, because \( 9 \times 9 = 81 \).

Thus, \( \sqrt{\frac{25}{81}} = \frac{5}{9} \). This process is crucial for simplifying expressions and solving equations.
By splitting the fraction under the square root into simpler parts, we can handle complex-looking math problems with ease.

Reducing Fractions

Reducing fractions is a fundamental skill in mathematics that involves simplifying fractions to their smallest form without changing their value. It's about making a fraction easier to understand by cutting it down to its smallest possible size.
To reduce a fraction, like \( \frac{15}{45} \), follow these steps:

• Find the greatest common divisor (GCD) of 15 and 45. The GCD is the largest number that divides both numbers without leaving a remainder. Here, the GCD is 15.
• Divide both the numerator (15) and the denominator (45) by their GCD. \( \frac{15 \div 15}{45 \div 15} = \frac{1}{3} \).

The result \( \frac{1}{3} \) is the simplest form of the original fraction \( \frac{15}{45} \). Reducing fractions is an essential step when solving fraction problems, as it makes them simpler to interpret and compute.

Lowest Terms

Expressing fractions in their lowest terms means writing them in the simplest form possible. In other words, the numerator and the denominator share no common factors other than 1.
Consider a fraction, such as \( \frac{15}{45} \). Although it seems complex, we can simplify it by dividing both the top number and the bottom number by their greatest common divisor:

• Identify the greatest common divisor of 15 and 45, which is 15.
• By dividing the numerator and denominator by this number: \( \frac{15 \div 15}{45 \div 15} = \frac{1}{3} \).

When in the lowest terms, a fraction like \( \frac{1}{3} \) is simpler to work with and understand.
Fractions in lowest terms are important in many areas of mathematics because they provide the clearest representation of the quantity they describe.