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Chapter 3: Problem 4

Simplify each expression. $$\frac{-7+7}{12-8}$$

Short Answer

Expert verified
The simplified expression is 0

Step by step solution

01

Perform Addition in the Numerator

First, perform the addition operation in the numerator. The sum of -7 and 7 is 0. Therefore, the new expression is \( \frac{0}{12-8}\).
02

Perform Subtraction in the Denominator

Next, perform the subtraction operation in the denominator. The result of 12 subtracted by 8 is 4. Therefore, the new expression is \( \frac{0}{4}\).
03

Simplify the Fraction

Finally, simplify the fraction. The numerator is 0 and any number (except for zero) divided by zero is 0. Therefore, the simplified expression is 0.

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Key Concepts

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

Understanding Numerator and Denominator
In fractions, the numerator is the top number, and the denominator is the bottom number. The numerator tells us how many parts we have, while the denominator tells us into how many equal parts the whole is divided.
This is crucial for performing any operations with fractions, including simplification. For instance, in the fraction \( \frac{-7+7}{12-8} \), \(-7+7\) is the numerator, and \(12-8\) is the denominator.
Understanding these positions helps us correctly apply mathematical operations to fractions.
Addition and Subtraction in Fractions
When dealing with fractions that involve addition or subtraction, it's essential to perform these operations separately in the numerator and the denominator.
In the expression \( \frac{-7+7}{12-8} \), we first handle the addition in the numerator: \(-7 + 7 = 0\).
This simplifies the expression's numerator to 0. Then, in the denominator, we perform the subtraction: \(12 - 8 = 4\).
This process is crucial for correctly setting up a fraction before simplification.
Fraction Simplification
Fraction simplification involves reducing a fraction to its simplest form. It means finding an equivalent fraction where the numerator and denominator cannot be reduced any further.
In our example, \( \frac{0}{4} \), the numerator is 0, making the entire fraction equal to 0.
  • Always check if the numerator is zero as it immediately simplifies the fraction to zero.
  • Another important note is that a fraction with 0 in the denominator is undefined, so be cautious of such cases.
Understanding when a fraction equals zero is critical for proper simplification.

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Most popular questions from this chapter

Express each variation as an equation. Then find the requested value. Assume that all variables represent positive numbers. \(y\) varies inversely with \(x^{2} .\) If \(y=6\) when \(x=4,\) find \(y\) when \(x=2\).

Set up a variation equation and solve for the requested value. The distance traveled by an object in free fall varies directly with the square of the time that it falls. If the object falls 256 feet in 4 seconds, how far will it fall in 6 seconds?

If points \(P(a, b)\) and \(Q(c, d)\) are two points on a rectangular coordinate system and point \(M\) is midway between them, then point \(M\) is called the midpoint of the line segment joining \(P\) and \(Q .\) (See the illustration on the following page. To find the coordinates of the midpoint \(M\left(x_{M}, y_{M}\right)\) of the segment PQ, we find the average of the \(x\) -coordinates and the average of the \(y\)-coordinates of \(P\) and \(Q\). $$x_{M}=\frac{a+c}{2}$$ and $$y_{M}=\frac{b+d}{2}$$ Find the coordinates of the midpoint of the line segment with the given endpoints. $$A(-2, y+1) \text { and } B(6, y-1)$$

Express each joint variation as an equation. Then find the requested value. Assume that all variables represent positive numbers. SEE EXAMPLE 3. (OBJECTIVE 3) D varies jointly with \(p\) and \(q\). If \(D=16\) when \(p\) and \(q\) are both 8 , find \(D\) when \(p\) and \(q\) are both 12 .

Graph each equation using any method. $$y=0$$
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