Problem 203 - Intermediate Algebra

Question

In the following exercises, simplify. $$ \frac{\left(\frac{3}{5}\right)^{2}}{\left(\frac{3}{7}\right)^{2}} $$

Step-by-Step Solution

Answer

\[ \frac{49}{25} \]

Step 1 - Simplify the numerator and the denominator

Simplify the squares in the numerator and the denominator: \ \[ \frac{\frac{9}{25}}{\frac{9}{49}} \]

Step 2 - Rewrite the division as a multiplication

Rewrite the division of fractions as multiplication by the reciprocal: \ \[ \frac{9}{25} \times \frac{49}{9} \]

Step 3 - Cancel out common factors

Cancel out the common factor (9) in the numerator and the denominator: \ \[ \frac{49}{25} \]

Step 4 - Write the final simplified fraction

The simplified result is: \ \[ \frac{49}{25} \]

Key Concepts

fraction division

Dividing fractions might seem tricky at first, but once you learn the steps, it becomes straightforward. To divide one fraction by another, you use a technique called 'multiplying by the reciprocal'. The reciprocal of a fraction is what you get when you flip the numerator and the denominator. For example, the reciprocal of \( \frac{3}{7} \) is \( \frac{7}{3} \). Instead of dividing by a fraction, you multiply by its reciprocal. This method makes the operation simpler and more intuitive.

fraction multiplication

Multiplying fractions is one of the simplest operations you can do with fractions. To multiply two fractions, you just multiply their numerators together and their denominators together. For instance, if you have \( \frac{a}{b} \) and \( \frac{c}{d} \), their product will be \( \frac{a \times c}{b \times d} \). This rule holds true no matter how complicated the fractions appear. Always remember to simplify the fraction in the end by finding the greatest common divisor (GCD) and dividing both the numerator and the denominator by this number.

simplification steps

Simplifying fractions involves reducing them to their simplest form. To do this, follow these steps:

• First, identify the greatest common divisor (GCD) of the numerator and the denominator.
• Next, divide both the numerator and the denominator by this GCD.
• If needed, perform any additional arithmetic operations required like canceling out common factors in the numerator and denominator.

By following these steps, you’ll reduce the fraction to its simplest form, making it easier to work with in subsequent calculations.