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Chapter 4: Problem 34

$$\text { Evaluate each expression if } x=-\frac{1}{3}, y=\frac{2}{5}, \text { and } z=\frac{5}{6} \text { . }$$ $$x^{2}-z^{2}$$

Short Answer

Expert verified
The expression evaluates to \(-\frac{7}{12}\).

Step by step solution

01

Substitute the given values into the expression

Start with the expression \(x^2 - z^2\). Substitute the values \(x = -\frac{1}{3}\) and \(z = \frac{5}{6}\) into the expression: \[\left(-\frac{1}{3}\right)^2 - \left(\frac{5}{6}\right)^2.\]
02

Calculate \(x^2\)

Determine the square of \(x = -\frac{1}{3}\):\[\left(-\frac{1}{3}\right)^2 = \frac{1}{9}.\]
03

Calculate \(z^2\)

Determine the square of \(z = \frac{5}{6}\):\[\left(\frac{5}{6}\right)^2 = \frac{25}{36}.\]
04

Subtract \(z^2\) from \(x^2\)

Subtract \(z^2\) from \(x^2\), finding the difference between the two fractions: \[\frac{1}{9} - \frac{25}{36}.\]
05

Find a common denominator

The common denominator of 9 and 36 is 36.
06

Convert \(\frac{1}{9}\) to have denominator 36

Convert \(\frac{1}{9}\) by making its denominator 36 so it can be subtracted from \(\frac{25}{36}\):\[\frac{1}{9} = \frac{4}{36}.\]
07

Perform the subtraction

Subtract the equivalent fractions: \[\frac{4}{36} - \frac{25}{36} = -\frac{21}{36}.\]
08

Simplify the fraction

Simplify \(-\frac{21}{36}\) by dividing both the numerator and the denominator by their greatest common divisor (3): \[-\frac{21}{36} = -\frac{7}{12}.\]

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Key Concepts

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

Squaring Fractions
Squaring a fraction involves multiplying the fraction by itself. This is an important step in our example exercise where we needed to square \(x = -\frac{1}{3}\) and \(z = \frac{5}{6}\).
  • When squaring a negative fraction like \(-\frac{1}{3}\), remember that the negative sign becomes positive. This is because the square of a negative number is always positive. So, \(\left(-\frac{1}{3}\right)^2 = \left(\frac{1}{3}\right)\times\left(\frac{1}{3}\right) = \frac{1}{9}\).
  • For \(z = \frac{5}{6}\), squaring it involves multiplying the fraction by itself: \(\left(\frac{5}{6}\right)^2 = \frac{5}{6}\times\frac{5}{6} = \frac{25}{36}\).
When squaring fractions, simply square both the numerator and the denominator. This simplifies your work and helps avoid common mistakes.
Common Denominator
A common denominator is essential when adding or subtracting fractions. It ensures that fractions have the same bottom number, allowing for the operation to be performed smoothly. In the expression \(\frac{1}{9} - \frac{25}{36}\), we need a common denominator to subtract these fractions effectively.
  • First, identify the least common multiple (LCM) of the denominators 9 and 36. In this scenario, 36 is the LCM.
  • Adjust the fractions so they both have this common denominator. The fraction \(\frac{1}{9}\) changes to \(\frac{4}{36}\) since \(1 \times 4 = 4\) and \(9 \times 4 = 36\).
Having a common denominator simplifies arithmetic operations on fractions, reducing complexity and minimizing potential errors.
Simplifying Fractions
After performing the operation on fractions, you may need to simplify the result. Simplifying a fraction means making it as easy to understand as possible by reducing it to its lowest terms.
  • In our exercise, the subtraction resulted in \(-\frac{21}{36}\).
  • To simplify, identify the greatest common divisor (GCD) of 21 and 36, which is 3. Divide both the numerator and denominator by this number.
  • Hence, \(-\frac{21}{36}\) simplifies to \(-\frac{7}{12}\), producing a cleaner and neater answer. Dividing by the GCD ensures you're not left with unnecessary or redundant terms.
Understanding how to simplify fractions is key to ensuring your final results are always presented clearly and concisely.

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Most popular questions from this chapter

Answer true or false for each statement. It is possible for the average of two numbers to be less than both numbers.

Perform the indicated operations. $$ -4 \frac{2}{5} \cdot 2 \frac{3}{10} $$

Write each mixed number as an improper fraction. See Example 20. $$9 \frac{7}{20}$$

Recall that to find the average of two numbers, we find their sum and divide by \(2 .\) For example, the average of \(\frac{1}{2}\) and \(\frac{3}{4}\) is \(\frac{\frac{1}{2}+\frac{3}{4}}{2}\). Find the average of each pair of numbers. Study Exercise \(67 .\) Without calculating, \(\operatorname{can} \frac{1}{3}\) be the average of \(\frac{1}{2}\) and \(\frac{8}{9}\) ? Explain why or why not.

Solve. A student asked you to check his work below. Is it correct? If not, where is the error? \(3 \frac{2}{3} \cdot 1 \frac{1}{7} \stackrel{?}{=} 3 \frac{2}{21}\)
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