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Chapter 5: Problem 12

Evaluate each expression. See Example 1. $$ \left(-\frac{1}{9}\right)^{2} $$

Short Answer

Expert verified
The expression evaluates to \( \frac{1}{81} \).

Step by step solution

01

Identify the Expression

We are given the expression \( \left(-\frac{1}{9}\right)^{2} \). This expression involves raising a fraction with a negative numerator to the power of 2.
02

Apply the Rules of Exponents

When you square a negative number, the result will be positive because multiplying two negative numbers results in a positive number. Therefore, \( \left(-\frac{1}{9}\right) \times \left(-\frac{1}{9}\right) \) will yield a positive result.
03

Multiply the Fractions

Compute \( \left(-\frac{1}{9}\right)^{2} \) by multiplying \( \left(-\frac{1}{9}\right) \times \left(-\frac{1}{9}\right) \). This involves multiplying the numerators and denominators separately:1. Multiply the numerators: - \( -1 \times -1 = 1 \)2. Multiply the denominators: - \( 9 \times 9 = 81 \)
04

Simplify the Result

The result of \( \left(-\frac{1}{9}\right)^{2} \) is \( \frac{1}{81} \). Since both the numerator and denominator are positive, \( \frac{1}{81} \) is already in its simplest form.

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Key Concepts

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

Understanding Fractions
Fractions are numbers that represent parts of a whole. They are composed of two parts:
  • Numerator: This is the top part of a fraction and indicates how many parts we have.
  • Denominator: This is the bottom part and shows the total number of equal parts the whole is divided into.
If we consider a fraction like \(-\frac{1}{9}\), the numerator is \(-1\), and the denominator is \(9\). This means that we have one part out of nine, but since the numerator is negative, it represents a subtraction or "loss" of that one part.
Negative fractions can be challenging because they combine two types of values: fractions and negatives. However, when handled carefully, they follow the same arithmetic rules as positive fractions.
Working with Negative Numbers
Negative numbers are numbers less than zero and are represented with a minus sign \(-\). They are used to express a deficit or a decrease.
When it comes to multiplication, they follow a unique rule: the product of two negative numbers is positive. So, if you multiply \(-1\) by itself, you get:
  • \(-1 \times -1 = 1\).
In our example, we applied this rule when squaring the fraction \(-\frac{1}{9}\). Since we multiply the negative numerator by itself, it results in a positive \(1\). Understanding this concept is crucial when dealing with exponentiation, which involves repeatedly multiplying a number by itself.
Simplifying Expressions
Simplifying expressions involves rewriting them in the most basic or concise form. In mathematics, this often means reducing fractions to their simplest terms or combining like terms.
When we simplified \( \left(-\frac{1}{9}\right)^{2} \), we followed these steps:
  • Calculated \(-1 \times -1\), resulting in \(1\).
  • Computed \(9 \times 9\) to get \(81\).
  • The final product of the expression is \(\frac{1}{81}\).
  • This fraction is already in its simplest form since there are no common factors between \(1\) and \(81\), other than \(1\).
The key to simplifying is patience and ensuring all steps follow mathematical principles, allowing you to reach the most reduced version of the expression.

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Most popular questions from this chapter

Use the distributive property and multiply.See Sections 5.3 and \(5.5 .\) $$ \frac{1}{y}\left(5 y^{2}-6 y+5\right) $$

Perform the indicated operation. $$\left(-7 m^{2}\right)^{2}$$

Multiply. $$\left(a^{2}+3 a-2\right)\left(2 a^{2}-5 a-1\right)$$

Multiply. $$\left(3 x^{2}+2 x-4\right)\left(2 x^{2}-4 x+3\right)$$

Simplify each expression. See Section \(5.1 .\) $$ \frac{-48 a b^{6}}{32 a b^{3}} $$
See all solutions

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