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Chapter 1: Problem 497

Simplify. $$ \frac{\left(\frac{3}{5}\right)^{2}}{\left(\frac{3}{7}\right)^{2}} $$

Short Answer

Expert verified
Simplified result is \ \ \frac{49}{25}.

Step by step solution

01

Simplify the Exponents

Simplify each exponent separately. \ \( \frac{3}{5} \) raised to the power of 2 is \( \frac{3^2}{5^2} \) which equals to \( \frac{9}{25} \). Similarly, \( \frac{3}{7} \) raised to the power of 2 is \( \frac{3^2}{7^2} \) which equals to \( \frac{9}{49} \).
02

Divide the Fractions

Divide the simplified fractions. To divide fractions, multiply by the reciprocal of the divisor. So, \( \frac{ \frac{9}{25} }{ \frac{9}{49} } = \frac{9}{25} \times \frac{49}{9} \).
03

Simplify the Multiplication

Simplify the multiplication by canceling out the common factors. \ \( \frac{9}{25} \times \frac{49}{9} = \frac{9 \times 49}{25 \times 9} \) and \ \ the 9's cancel out, giving us \ \( \frac{49}{25} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponents
Exponents are used to express repeated multiplication of the same number. For example, if we have \(3^2\), it means 3 multiplied by itself, which equals 9. The same goes for fractions. To simplify an expression like \(\left(\frac{3}{5}\right)^2\), you need to apply the exponent to both the numerator and the denominator separately. So, \(\left(\frac{3}{5}\right)^2 = \frac{3^2}{5^2} = \frac{9}{25}\). Understanding how to handle exponents on fractions is critical for simplifying complex fractional expressions.
Fractions
Fractions represent a part of a whole and are written as \(\frac{a}{b}\), where 'a' is the numerator and 'b' is the denominator. Fractions are all around us, from simple recipes to complex mathematical problems. Simplifying fractions involves reducing them to their simplest form by dividing both the numerator and the denominator by their greatest common divisor (GCD). For example, the fraction \(\frac{9}{25}\) is already in its simplest form since 9 and 25 cannot be divided by a common number other than 1.
Reciprocal
The reciprocal of a fraction is simply flipping the numerator and the denominator. If you have \(\frac{a}{b}\), the reciprocal is \(\frac{b}{a}\). This concept is particularly useful in division, especially when working with fractions. Instead of dividing by a fraction, you multiply by its reciprocal. Take \(\frac{9}{25} \div \frac{9}{49}\): instead of dividing, you multiply \(\frac{9}{25}\) by the reciprocal of \(\frac{9}{49}\), which is \(\frac{49}{9}\). This makes computations easier and avoids the tricky process of direct division involving fractions.
Multiplication of Fractions
Multiplying fractions involves multiplying their numerators together and their denominators together. For example, \(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\). This operation is relatively straightforward. However, you often need to simplify the result by canceling out common factors. In our exercise, after multiplying \(\frac{9}{25} \times \frac{49}{9}\), we get \(\frac{9 \times 49}{25 \times 9}\). The 9's cancel out, leaving us with \(\frac{49}{25}\). Simplifying the final result is crucial to ensure your answer is as neat and readable as possible.

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