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Chapter 10: Problem 572

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

Short Answer

Expert verified
\( \frac{9}{25} \)

Step by step solution

01

Understand the Power Rule

The expression \( \left(\frac{3}{5}\right)^{2} \) involves raising the fraction \(\frac{3}{5}\) to the power of 2. This means both the numerator and the denominator of the fraction need to be squared.
02

Square the Numerator

Square the numerator of the fraction: \((3)^{2} = 9\).
03

Square the Denominator

Square the denominator of the fraction: \((5)^{2} = 25\).
04

Form the Simplified Fraction

Combine the results from Steps 2 and 3 to form the simplified fraction: \( \frac{9}{25} \).

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Key Concepts

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

Power Rule: Fundamental in Exponents
The Power Rule is a fundamental concept in working with exponents. It states that when you raise a quotient (or fraction) to a power, you need to apply the exponent to both the numerator and the denominator separately. For instance, when you have an expression like \(\bigg(\frac{a}{b}\bigg)^n\), you should calculate \(a^n\) and \(b^n\) individually.
Let's break it down step by step:
  • Identify the fraction: \(\frac{3}{5}\)
  • Apply the exponent to both parts: \( \bigg(\frac{3}{5}\bigg)^2 = \frac{3^2}{5^2} \)
  • Simplify the result: \( \frac{9}{25} \)
This method ensures that every part of the fraction is accurately raised to the intended power.
Squaring Fractions: Doubling the Power
Next, let's talk about squaring fractions, which is an extension of the power rule. To square a fraction, you multiply the fraction by itself, which also means you raise both the numerator and the denominator to the power of two.
Consider the fraction \(\frac{3}{5}\):
  • Square the numerator: \( (3)^2 = 9 \)
  • Square the denominator: \( (5)^2 = 25 \)
  • Form the new fraction: \(\frac{9}{25}\)
Every fraction squared will follow this rule. Multiplying the fraction by itself achieves the same result. Just remember that both components - the top number and the bottom number - are raised to the power of two.
Simplifying Expressions: Making Math Manageable
Simplifying expressions is an essential skill in mathematics, enabling you to reduce complex expressions to a simpler form. When simplifying, we often look to apply rules like the power rule or operations like squaring.
Consider simplifying \(\bigg(\frac{3}{5}\bigg)^2\):
  • Apply the power rule by squaring both the numerator and the denominator.
  • Square the numerator: \( (3)^2 = 9 \).
  • Square the denominator: \( (5)^2 = 25 \).
  • Combine them into a simplified fraction: \(\frac{9}{25}\).
Another aspect to pay attention to is reducing fractions to their simplest form. However, in this case, \(\frac{9}{25}\) is already in its simplest form, as the numerator and the denominator do not have any common factors other than 1. Simplifying expressions like this helps make more challenging problems easier to manage and solve.

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