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Chapter 5: Problem 22

Evaluate the expression using the given values. \(m v^{2} ; m=6 \frac{1}{2}, v=-\frac{2}{5}\)

Short Answer

Expert verified
Question: Evaluate the expression \(m v^{2}\) for \(m=6\frac{1}{2}\) and \(v=-\frac{2}{5}\). Answer: \(\frac{26}{25}\)

Step by step solution

01

Insert the given values into the expression

Replace \(m\) with \(6\frac{1}{2}\) and \(v\) with \(-\frac{2}{5}\) in the expression \(m v^{2}\). This gives us \((6\frac{1}{2})(-\frac{2}{5})^2\).
02

Convert the mixed number to an improper fraction

Convert \(6\frac{1}{2}\) into an improper fraction. Since \(6\frac{1}{2}=6+1/2\), we can write it as \(6*2/2+1/2\), which equals \(\frac{12+1}{2}\). Therefore, \(6\frac{1}{2} = \frac{13}{2}\). Now our expression becomes: \((\frac{13}{2})(-\frac{2}{5})^2\)
03

Square the velocity term

Square the given value of \(v\), which is \(-\frac{2}{5}\). We have \((-\frac{2}{5})^2 = \frac{(-2)^2}{5^2} = \frac{4}{25}\). Now our expression becomes: \((\frac{13}{2})(\frac{4}{25})\)
04

Multiply the fractions

Multiply the two fractions as follows: \(\frac{13}{2} * \frac{4}{25} = \frac{13*4}{2*25} = \frac{52}{50}\).
05

Simplify the fraction

Divide the numerator and the denominator of the obtained fraction by their greatest common divisor (GCD). The GCD of \(52\) and \(50\) is \(2\). So, \(\frac{52}{50} = \frac{52/2}{50/2} = \frac{26}{25}\). Hence, the evaluated expression is \(\frac{26}{25}\).

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Key Concepts

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

Fractions
Fractions are a way to represent parts of a whole. They consist of a numerator and a denominator separated by a line. For example, in the fraction \(\frac{3}{4}\), \(3\) is the numerator indicating how many parts we have, and \(4\) is the denominator showing the total number of equal parts.

When working with fractions, there are several key operations:
  • **Addition/Subtraction**: Fractions must have the same denominator to be added or subtracted. For example, \(\frac{1}{2} + \frac{1}{3}\) requires finding a common denominator, like \(6\), resulting in \(\frac{3}{6} + \frac{2}{6} = \frac{5}{6}\).
  • **Multiplication**: Simply multiply the numerators and denominators. For example, \(\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}\).
  • **Division**: Flip the second fraction and multiply. For instance, \(\frac{2}{3} \div \frac{3}{4} = \frac{2}{3} \times \frac{4}{3} = \frac{8}{9}\).
Fractions can often be simplified. This means dividing the numerator and the denominator by their greatest common divisor (GCD). Simplifying \(\frac{24}{36}\) by their GCD, \(12\), gives \(\frac{2}{3}\).
Understanding how to manipulate fractions is crucial for evaluating expressions.
Mixed Numbers
Mixed numbers consist of a whole number and a fraction. For example, \(6\frac{1}{2}\) includes the whole number \(6\) and the fraction \(\frac{1}{2}\). They're a convenient way to express numbers greater than one.

To use them in mathematical calculations, it's often necessary to convert them to improper fractions:
  • **Convert the Whole Number**: Multiply the whole number by the denominator of the fraction part.
  • **Add the Fraction**: Add this product to the numerator of the fraction part.
  • **Create Fraction**: Use the sum as the new numerator, with the original denominator remaining the same.
For example, to convert \(6\frac{1}{2}\) to an improper fraction:
  • Multiply \(6\) by \(2\): \(6 \times 2 = 12\)
  • Add the numerator \(1\): \(12 + 1 = 13\)
  • Write the fraction \(\frac{13}{2}\)
Converting mixed numbers to improper fractions enables operations like multiplication and division more easily in expressions.
Squaring Numbers
Squaring a number means multiplying it by itself. It's a fundamental operation in algebra and often used in various equations and expressions. In mathematical terms, squaring \(a\) is represented as \(a^2\).

When squaring fractions, the process remains the same:
  • **Numerator**: Square the numerator.
  • **Denominator**: Square the denominator.
  • **Result**: Keep the result as a new fraction.
For instance, squaring the fraction \(-\frac{2}{5}\):
  • Square \(-2\): \((-2)^2 = 4\)
  • Square \(5\): \(5^2 = 25\)
  • Result: \(\frac{4}{25}\)
Squaring retains the sign when dealing with negative numbers because multiplying two negative numbers results in a positive number. This concept is pivotal when evaluating expressions involving exponents.

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Most popular questions from this chapter

Solve. Heather's grade in a course is determined by the average (mean) of six tests. Her scores on the first five tests are \(83,78,82,70,\) and \(76 .\) To get a \(\mathrm{B}\), she must have a final average of 80 or higher. What is the minimum score she needs on the sixth test to receive a B?

Solve. A tabletop is \(\frac{3}{4}\) inch thick. During sanding, \(\frac{1}{16}\) inch is removed off the top. How thick is the tabletop after sanding?

Add or subtract. $$-7 \frac{3}{4}-\left(-1 \frac{1}{8}\right)$$

Add or subtract. $$6 \frac{3}{4}+\left(-2 \frac{1}{3}\right)$$

Estimate and find the actual sum expressed as a mixed mumber in simplest form. $$8 \frac{3}{7}-2 \frac{1}{7}$$
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