Problem 97 - Fundamentals of Mathematics

Question

For the following problems, find the products. Be sure to reduce. $$\left(\frac{3}{7}\right)^{2} \cdot\left(\frac{1}{9}\right)^{2}$$

Step-by-Step Solution

Answer

The product is \(\frac{1}{441}\).

Step 1: Calculate the Squares

First, calculate the square of each fraction separately. \((\frac{3}{7})^2\) means multiplying \(\frac{3}{7}\) by itself. This gives \(\frac{9}{49}\). Similarly, \((\frac{1}{9})^2\) means multiplying \(\frac{1}{9}\) by itself, which equals \(\frac{1}{81}\).

Step 2: Multiply the Fractions

Next, multiply the results of the squares together. Multiply the numerators: \(9 \times 1 = 9\). Multiply the denominators: \(49 \times 81 = 3969\). This gives \(\frac{9}{3969}\).

Step 3: Simplify the Fraction

Finally, simplify \(\frac{9}{3969}\). Both 9 and 3969 are divisible by 9. Dividing the numerator and the denominator by 9 results in \(\frac{1}{441}\). Since 441 is 21 squared, it cannot be reduced any further, this is the simplest form.

Key Concepts

Simplifying Fractions

Simplifying fractions involves reducing the fraction to its simplest form, which means both the numerator and the denominator are divided by their greatest common divisor (GCD). In our exercise, after calculating the product of the squares, we end up with the fraction \(\frac{9}{3969}\). Both 9 and 3969 are divisible by 9, their GCD, so we divide each by 9:

• The new numerator is \(\frac{9}{9} = 1\)
• The new denominator is \(\frac{3969}{9} = 441\)

Therefore, \(\frac{9}{3969}\) simplifies to \(\frac{1}{441}\). Simplifying fractions like this helps make them easier to understand and work with in further calculations.

Exponents

Exponents represent repeated multiplication of the same number. When we see \((\frac{3}{7})^2\), this means \(\frac{3}{7}\) is multiplied by itself — essentially \(\frac{3}{7} \times \frac{3}{7}\).
For any fraction \(\left(\frac{a}{b}\right)^n\), you compute it as:

• Numerator: \(a^n\)
• Denominator: \(b^n\)

This is how \(\left(\frac{1}{9}\right)^2 = \frac{1}{81}\) is derived, by multiplying the numerator and the denominator by 1 and 9 respectively.
Exponents simplify repeated multiplications into a more compact form, making calculations more efficient and manageable.

Numerator and Denominator

The terms numerator and denominator are fundamental to understanding fractions. The numerator is the top number in a fraction, representing how many parts we have. The denominator is the bottom number, showing the total number of equal parts in a whole.
In the given problem \(\left(\frac{3}{7}\right)^2 \cdot \left(\frac{1}{9}\right)^2\), for each squared fraction:

• The first fraction \(\frac{3}{7}\) has a numerator of 3 and a denominator of 7.
• The second fraction \(\frac{1}{9}\) has a numerator of 1 and a denominator of 9.

After squaring, these become \(\frac{9}{49}\) and \(\frac{1}{81}\), respectively. Knowing which number is the numerator and which is the denominator is crucial when multiplying or reducing fractions.

Squared Fractions

When we square a fraction, we apply the exponent to both the numerator and the denominator separately. This means for a fraction \(\left(\frac{a}{b}\right)^2\), the operation is \(a^2\) for the numerator and \(b^2\) for the denominator.
In our exercise:

• Squaring \(\frac{3}{7}\) results in \(\frac{9}{49}\) because \(3^2 = 9\) and \(7^2 = 49\).
• Squaring \(\frac{1}{9}\) results in \(\frac{1}{81}\) because \(1^2 = 1\) and \(9^2 = 81\).

Squaring helps in situations where repeated multiplication is needed, especially in geometric problems or when dealing with areas in mathematics. This operation becomes straightforward when you recognize the pattern of applying the square to both parts of the fraction separately.