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Chapter 4: Problem 2

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

Short Answer

Expert verified
\( \frac{1}{81} \)

Step by step solution

01

Understand the Expression

The given expression is \(\bigg(\frac{1}{9}\bigg)^{2}\). It represents raising the fraction \( \frac{1}{9} \) to the power of 2.
02

Apply the Power Rule

Using the power rule for fractions, \( ( \frac{a}{b} )^n = \frac{a^n}{b^n} \), raise both the numerator and the denominator to the power of 2: \(\bigg(\frac{1}{9}\bigg)^{2} = \frac{1^{2}}{9^{2}}\).
03

Simplify the Numerator and Denominator

Calculate the powers: \(1^{2} = 1\) and \(9^{2} = 81\). This results in \( \frac{1}{81} \).

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Key Concepts

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

Power Rule for Fractions
When you encounter a fraction raised to a power, you need to apply the power rule for fractions. This rule states that \(( \frac{a}{b} )^n = \frac{a^n}{b^n}\). Essentially, you raise both the numerator and the denominator to the given power separately.
As an example, consider the fraction \(\frac{1}{9}\) raised to the power of 2. Using the power rule, you get \(\bigg(\frac{1}{9}\bigg)^{2} = \frac{1^{2}}{9^{2}}\). This simplifies the problem and allows you to work with smaller pieces of the fraction independently.
Simplifying Exponents
The next step involves simplifying the exponents in the numerator and the denominator. Each part of the fraction, the numerator and the denominator, must be raised to the given power.
For our example, calculate \(\frac{1^{2}}{9^{2}}\). Begin by tackling the numerator, \(1^{2} = 1\). Then, handle the denominator, \(9^{2} = 81\). Simplifying these components separately makes the process more manageable.
After applying these operations, the expression becomes \(\frac{1}{81}\).
Numerator and Denominator
Understanding fractions requires a solid grasp of both numerators and denominators. The numerator is the top number of the fraction, indicating how many parts we have. The denominator is the bottom number, showing the total number of parts.
For \(\frac{1}{9}\), 1 is the numerator, and 9 is the denominator. Raising each to the power of 2 means working separately on these components: multiplying the numerator by itself and the denominator by itself.
This keeps calculations straightforward and leads us to our final simplified result, \(\frac{1}{81}\). Remember, accurate manipulation of the numerator and denominator is key to successfully simplifying fractions.

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Most popular questions from this chapter

Simplify to lowest terms by first reducing the powers of 10. $$-\frac{2000}{1500}$$

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