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Chapter 4: Problem 9

Simplify. $$\left(\frac{1}{9}\right)^{2}$$

Short Answer

Expert verified
\[ \left(\frac{1}{9}\right)^{2} = \frac{1}{81} \]

Step by step solution

01

- Understand the Problem

You are asked to simplify the expression \(\left( \frac{1}{9} \right)^{2}\).
02

- Apply Exponent Rule

Recall the rule for exponents: \(\left( \frac{a}{b} \right)^{n} = \frac{a^{n}}{b^{n}}\). This means you need to apply the exponent to both the numerator and the denominator.
03

- Simplify the Numerator

In this case, the numerator is 1, so \(1^{2} = 1\).
04

- Simplify the Denominator

The denominator is 9, so \(9^{2} = 81\).
05

- Combine Results

Put the simplified numerator and denominator together to get the final answer: \(\frac{1}{81}\).

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Key Concepts

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

Exponent Rules
Exponent rules help us to simplify expressions involving powers. These rules dictate how we handle exponents in different mathematical operations. One important rule is \( \left( \frac{a}{b} \right)^{n} = \frac{a^{n}}{b^{n}}\). This means the exponent applies to both the numerator and the denominator. Let's look at our example \( \left( \frac{1}{9} \right)^{2}\). Here, the exponent 2 applies to both 1 and 9. Other common rules include \(a^{m} \times a^{n} = a^{m+n}\) and \( \frac{a^{m}}{a^{n}} = a^{m-n} \).
Numerator and Denominator
In a fraction, the numerator is the top number, and the denominator is the bottom number. Understanding the roles of these parts is crucial for simplifying fractions with exponents. In our example \( \left( \frac{1}{9} \right)^{2}\), the numerator is 1 and the denominator is 9. According to the exponent rule we discussed, the exponent 2 is applied to both the numerator and the denominator:
Fractions
Fractions represent a part of a whole. They consist of a numerator and a denominator. Simplifying fractions with exponents involves breaking the problem down using exponent rules. For \( \left( \frac{1}{9} \right)^{2} \), we apply the exponent to both the numerator and denominator separately. This simplifies to \( \frac{1^{2}}{9^{2}} = \frac{1}{81}\). Thus, simplifying fractions using exponent rules gives us a more straightforward fraction.

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