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Chapter 4: Problem 49

Simplify the following. $$\left(-\frac{2}{9}-\frac{7}{9}\right)^{4}$$

Short Answer

Expert verified
The simplified expression is 1.

Step by step solution

01

Simplify the Expression Inside the Parentheses

First, simplify the expression inside the parentheses: \(-\frac{2}{9} - \frac{7}{9}\). Since these fractions have the same denominator, you can subtract the numerators directly.\[-\frac{2}{9} - \frac{7}{9} = \frac{-2 - 7}{9} = \frac{-9}{9} = -1\]
02

Apply the Exponent to the Simplified Expression

Now that the expression inside the parentheses simplifies to \(-1\), apply the exponent:\[(-1)^4\]Since \(-1\) raised to an even power results in 1, calculate:\[(-1)^4 = 1\]
03

Conclude the Simplification

The entire expression simplifies to 1. Thus, the simplified form of the original expression \(\left(-\frac{2}{9}-\frac{7}{9}\right)^{4}\) is 1.

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Key Concepts

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

Fraction Subtraction
Subtracting fractions is a fundamental part of pre-algebra you will find essential throughout math. To subtract fractions successfully, ensure they have a common denominator. With a shared denominator, it's a straightforward operation.
\(-\frac{2}{9} - \frac{7}{9}\) presents a perfect opportunity to practice this. Here, the denominators are already the same: 9.
  • Focus on subtracting the numerators by simply calculating \(-2 - 7\).
  • This sums to \(-9\).
  • The denominator remains 9, as it is common between both fractions.
Thus, the expression simplifies to \(\frac{-9}{9}\). Recognizing this as \(-1\) completes the subtraction step with ease.
Exponentiation
Exponentiation involves raising a number, called the base, to a power, called the exponent. In the context of our given expression, we dealt with the base of \(-1\) and the exponent 4. It's very important to understand the effect exponents have, especially on negative bases.
  • Here, we need to calculate \((-1)^4\).
  • Raising \(-1\) to an even power (like 2, 4, 6, etc.) always results in a positive 1. This is because:
    • \((-1)^2 = 1\)
    • \((-1)^4 = 1\)
    • Continues similarly for higher even exponents.
This property makes simplifying such expressions predictable. So, in this exercise, \((-1)^4 = 1\).
Negative Numbers
When dealing with negative numbers, getting comfortable with their addition, subtraction, and exponentiation is crucial. In this exercise, we encountered negativity both in the original fractional terms and as the base for the power.
  • Adding or subtracting negative numbers can impact your results. For instance:
    • The subtraction \(-2 - 7 = -9\) could intuitively be seen as moving further left on a number line, deepening into negative values.
  • Even powers of \(-1\) flip the sign non-effectually, ensuring positivity. Odd ones, however, keep it negative.
    • e.g., \((-1)^1 = -1\), \((-1)^3 = -1\)
This pattern provides essential foundational understanding as you advance through your arithmetic journey.

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Most popular questions from this chapter

Simplify the following. $$\frac{\frac{x}{3}+2}{5+\frac{1}{3}}$$

$$\text { Evaluate each expression if } x=\frac{3}{4} \text { and } y=-\frac{4}{7}$$ $$x^{2}+7 y$$

Perform each indicated operation. If the result is an improper fraction, also write the improper fraction as a mixed number. $$4-\frac{1}{5}$$

Write each fraction in two other equivalent ways by inserting the negative sign in different places. $$-\frac{11}{2}=\quad=$$

In your own words, explain how to find the \(L C D\) of two fractions.
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