Problem 106 - Fundamentals of Mathematics

Question

For the following problems, find each value. Reduce answers to lowest terms or convert to mixed numbers. $$\left(1 \frac{3}{4}\right)^{2} \cdot \sqrt{\frac{4}{49}}$$

Step-by-Step Solution

Answer

The value is \(\frac{7}{8}\).

Step 1: Convert Mixed Number to Improper Fraction

Convert the mixed number \(1 \frac{3}{4}\) into an improper fraction. To do this, multiply the whole number 1 by the denominator 4 and add the numerator 3: \(1\times4 + 3 = 7\). Therefore, the improper fraction is \(\frac{7}{4}\).

Step 2: Square the Improper Fraction

Square the improper fraction \(\frac{7}{4}\). To do this, square the numerator and the denominator: \(\left(\frac{7}{4}\right)^2 = \frac{7^2}{4^2} = \frac{49}{16}\).

Step 3: Simplify the Square Root

Calculate the square root of \(\frac{4}{49}\). The square root of a fraction is the square root of the numerator divided by the square root of the denominator: \(\sqrt{\frac{4}{49}} = \frac{\sqrt{4}}{\sqrt{49}} = \frac{2}{7}\).

Step 4: Multiply the Results

Multiply the squared fraction \(\frac{49}{16}\) by the simplified square root \(\frac{2}{7}\): \(\frac{49}{16} \times \frac{2}{7} = \frac{49 \times 2}{16 \times 7} = \frac{98}{112}\).

Step 5: Reduce the Fraction to Lowest Terms

Reduce \(\frac{98}{112}\) to its lowest terms by finding the greatest common divisor (GCD) of 98 and 112. The GCD is 14. Divide both the numerator and denominator by 14: \(\frac{98\div14}{112\div14} = \frac{7}{8}\).

Key Concepts

Improper Fractions

Understanding improper fractions is a crucial step when solving problems involving fractions. An improper fraction is defined as a fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number). For example, \(\frac{7}{4}\) is an improper fraction since 7 is greater than 4.
These types of fractions often appear when converting mixed numbers or performing mathematical operations involving fractions. To convert a mixed number into an improper fraction:

• Multiply the whole number by the denominator.
• Add the result to the numerator.
• Place that sum over the original denominator.

By understanding these concepts, you can work seamlessly between different types of fractions.

Mixed Numbers

Mixed numbers combine a whole number and a fraction into a single quantity. For instance, 1 \(\frac{3}{4}\) is a mixed number that represents the sum of 1 and \(\frac{3}{4}\). These forms are useful for visualizing quantities greater than one without resorting to improper fractions.
To convert a mixed number to an improper fraction, follow these steps:

• Multiply the whole number by the denominator of the fractional part.
• Add the numerator to this product.
• The result becomes the new numerator with the original denominator.

Converting mixed numbers to improper fractions often simplifies mathematical operations.

Square Root

The square root operation is finding a value that, when multiplied by itself, gives the original number. When dealing with fractions, the square root is applied to both the numerator and the denominator separately.
For the fraction \(\frac{4}{49}\), the square roots can be calculated as follows:

• The square root of 4 is 2, because 2 \(\times\) 2 = 4.
• The square root of 49 is 7, because 7 \(\times\) 7 = 49.

Thus, \(\sqrt{\frac{4}{49}} = \frac{2}{7}\). Understanding square roots of fractions allows you to simplify expressions by reducing complexity.

Reduce Fractions

Reducing fractions involves simplifying a fraction to its lowest terms. This process makes the fraction easier to understand and work with, ensuring expressions are neat and algebraically simple.
To reduce \(\frac{98}{112}\) to its simplest form, you need to calculate the greatest common divisor (GCD):

• Identify common factors of the numerator and the denominator.
• Divide both the numerator and the denominator by their GCD.

The GCD of 98 and 112 is 14. By dividing, you simplify to \(\frac{7}{8}\). Simplifying fractions is an essential skill in math, helping to streamline calculations and present results clearly.