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Chapter 7: Problem 64

Simplify each of the following as completely as possible. $$\frac{(3-5-2-6)^{2}}{(-1-2-3-4)^{2}}$$

Short Answer

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Step by step solution

01

Simplify the numerator and denominator separately

First, simplify the expression inside the parentheses in the numerator: \(3 - 5 - 2 - 6\). Then, do the same for the denominator: \(-1 - 2 - 3 - 4\).
02

Perform the arithmetic operations in the numerator

Calculate the value inside the numerator parentheses: \(3 - 5 - 2 - 6 = -10\).
03

Perform the arithmetic operations in the denominator

Calculate the value inside the denominator parentheses: \(-1 - 2 - 3 - 4 = -10\).
04

Square the simplified values

Now square the simplified values of the numerator and the denominator: \((-10)^{2}\) and \((-10)^{2}\).
05

Simplify the squared values

Calculate the squared values: \((-10)^{2} = 100\).
06

Divide the squared values

Finally, divide the squared value of the numerator by the squared value of the denominator: \(\frac{100}{100} = 1\).

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Key Concepts

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

Arithmetic Operations
When dealing with arithmetic operations, remember the basics:
  • Addition (+)
  • Subtraction (-)
  • Multiplication (脳)
  • Division (梅)
For this problem, we primarily use subtraction.
Start by simplifying within the parentheses:
In the numerator, subtract each number consecutively: \(3 - 5 - 2 - 6\).
Similarly, do so for the denominator: \(-1 - 2 - 3 - 4\).
Each subtraction involves reducing the previous result with the next number in line.
Practicing your subtraction helps you get quicker and more accurate.
This is a fundamental skill in algebraic simplification.
Numerator and Denominator
In any fraction, the number on top is the numerator, and the number on the bottom is the denominator.
Numerator: The part above the fraction line.
Denominator: The part below the fraction line.
To simplify our fraction:
\(\frac{(3-5-2-6)^2}{(-1-2-3-4)^2}\)
First, simplify the contents of both the numerator and the denominator:
Numerator: \(3 - 5 - 2 - 6 = -10\).
Denominator: \(-1 - 2 - 3 - 4 = -10\).
Once simplified, we square the results of the numerator and denominator separately.
Understanding numerator and denominator makes it easier to see how changes affect the whole fraction.
Squaring Numbers
Squaring a number means multiplying it by itself.
Mathematically, if you have a number \(x\), its square is \(x^2\).
In this exercise, we square -10 for both the numerator and the denominator:
Numerator: \((-10)^2 = 100\).
Denominator: \((-10)^2 = 100\).
Squaring always results in a positive number:
no matter if you start with a negative number!
After squaring numbers, we handle the simplified values:
our final fraction becomes \(\frac{100}{100}\).
When the numerator equals the denominator, the fraction simplifies to 1.

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Most popular questions from this chapter

Do your computation using scientific notation. $$\frac{(480)(0.008)}{(0.24)(4000)}$$

Suppose that the price \(p\) of a given item is related to the number of items sold \(x\) by the equation \(p=50-0.004 x\). The revenue \(R\) is computed by multiplying the price per item \(p\) by the number of items \(x\). Thus, \(R=x(50-0.004 x)\). Find the increase in revenue if \(x\) is increased by 500 .

Perform the indicated operations and simplify. $$\frac{4 x^{2} y}{15 z^{3}} \cdot \frac{12 x z}{x^{2} y}$$

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