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Chapter 7: Problem 55

Simplify each of the following as completely as possible. $$\frac{-u^{3}}{(-u)^{3}}$$

Short Answer

Expert verified
The simplified expression is 1.

Step by step solution

01

Understand the problem

We need to simplify the expression \(\frac{-u^{3}}{(-u)^{3}}\). To do this, we'll work with the powers and handle the negative signs accordingly.
02

Simplify the denominator

Notice that \((-u)^{3} = -(u^{3})\). This is because raising a negative number to an odd power results in a negative outcome. So the expression can be rewritten as \(\frac{-u^{3}}{-(u^{3})}\).
03

Cancel common factors

Now, we have \(\frac{-u^{3}}{-(u^{3})}\), where both the numerator and the denominator have the same term \(-u^{3}\). Negative signs cancel each other out: \(\frac{-u^{3}}{-u^{3}} = 1\).

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Key Concepts

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

Fraction Simplification
Fraction simplification is the process of making a fraction as simple as possible. The goal is to ensure both the numerator and the denominator have no common factors other than 1. You simplify a fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD). Here鈥檚 an example: To simplify the fraction \(\frac{4}{8}\), you find the GCD of 4 and 8, which is 4, then divide both terms by 4 to get \(\frac{1}{2}\).
In our exercise, we took \(\frac{-u^{3}}{(-u)^{3}}\), and simplified it. Initially, simplifying was complex due to the negative sign.
Powers and Exponents
Powers and exponents are mathematical shorthand for repeated multiplication. For instance, \(\text{3 to the power of 4}\) means multiplying 3 by itself 4 times, written as \(\text{3}^{4} = 3 \times 3 \times 3 \times 3\). The base number 3 is raised to the exponent 4.
The rules for exponents make simplifying expressions simpler. Key rules include:
  • \(\text{a}^{m} \times \text{a}^{n} = \text{a}^{m+n} \)
  • \(\frac{\text{a}^{m}}{\text{a}^{n}} = \text{a}^{m-n} \)
  • \((\text{a}^{m})^{n} = \text{a}^{m \times n} \)
  • Negative exponents: \(\text{a}^{-n} = \frac{1}{\text{a}^{n}} \)

In the exercise, we used exponents' properties to simplify the denominator, converting \((-u)^{3}\) to \(-(u^3)\).
Negative Numbers
Negative numbers are less than zero, represented with a minus sign (-). Working with negatives can be tricky, but some basic rules help. For multiplication and division:
  • A negative times a positive is a negative (-3 x 2 = -6).
  • A negative times a negative is a positive (-3 x -2 = 6).
  • Similarly, dividing follows the same rules: negative divided by positive remains negative, while negative divided by negative results in positive.

In the given problem, we handled negative \((-u)^{3}\) by recognizing that raising a negative number to an odd power yields a negative result. This property allowed us to ultimately simplify \(\frac{-u^{3}}{-(u^3)} = 1\), as the negative signs cancel each other out.

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Most popular questions from this chapter

Multiply out each of the following. As you work out the problems, identify those exercises that are either a perfect square or the difference of two squares. $$(2 a-5)^{2}$$

Given the expressions \((x y)^{2}\) and \((x+y)^{2},\) explain: (a) How are the two expressions similar? (b) How are they different? (c) How should each one be multiplied out?

Multiply out each of the following. As you work out the problems, identify those exercises that are either a perfect square or the difference of two squares. $$(3 x-4)(x+7)$$

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Multiply out each of the following. As you work out the problems, identify those exercises that are either a perfect square or the difference of two squares. $$(x-4)\left(x^{2}+6 x-7\right)$$
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