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Chapter 2: Problem 29

Find the product. $$(-13)(-2)(-2)\left(-\frac{2}{13}\right)$$

Short Answer

Expert verified
The product is -8.

Step by step solution

01

Multiply the integers

Begin by multiplying the integers \( -13 \), \( -2 \), and \( -2 \): \[ (-13)(-2)(-2) = 52 \]. Here, the product of three negative numbers is negative.
02

Multiply with the fraction

Now, multiply this result by the fraction \( -\frac{2}{13} \): \[ 52(-\frac{2}{13}) = -8 \]. Here, the product of a positive number and a negative number is negative. This concludes the solution.

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Key Concepts

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

Understanding Negative Numbers
Negative numbers are numbers less than zero, represented with a "-" sign. They can be a bit tricky when it comes to multiplication. Here's a simple way to think about them:
  • Whenever you multiply two numbers and one of them is negative, the result is negative.
  • If you multiply two negative numbers, the negatives "cancel out," making the product positive.
  • For example, multiplying \(-3\) and \(-3\), or \(-3 \times -3\), results in \(9\).
When you multiply multiple negative numbers, like in the task \((-13)(-2)(-2)\), start by multiplying two at a time:
  • The product of the first two, \((-13 \times -2) = 26\), is positive.
  • Then multiply the result by the third, \((26 \times -2) = -52\), which turns it back to negative.

Recognizing these patterns can make working with negative numbers less daunting.
Working with Fractions
Fractions represent parts of a whole and are written as the ratio of two numbers, with a numerator above and a denominator below a dividing line. They are important in many areas of math, including multiplication. Here's how you can easily deal with fractions in multiplication:
  • To multiply a whole number by a fraction, you can turn the whole number into a fraction by placing it over 1. For example, \(52\) can be written as \(\frac{52}{1}\).
  • Next, multiply the numerators and denominators across. So, for \(52 \times \left(-\frac{2}{13}\right)\), calculate \(\frac{52 \times -2}{1 \times 13} = \frac{-104}{13}\).
  • The division of \(-104\) by \(13\) gives \(-8\), as seen in the final step from the solution.

Knowing how to convert and multiply with fractions can simplify complex expressions.
Mastering Multiplying Integers
Multiplying integers involves considering both the size and the sign (positive or negative) of the numbers involved. Here are some tips to master it:
  • Identify the type (positive or negative) of each integer you're working with.
  • Recall that the product of two integers with the same sign is positive, and with different signs is negative.
  • In sequential multiplication (more than two numbers), pair them off and perform multiplication step by step to keep from getting confused.
  • For an odd number of negative integers, the overall product is negative. For an even number, it is positive.

For the exercise, combining negative integers such as \(-13\), \(-2\), and another \(-2\), you follow these steps iteratively:
- First, \(-13\times -2 = 26\) (positive).
- Then, \(26 \times -2 = -52\) (negative).
- Finally, multiply this result by the fraction, resulting in \(-8\).
Approaching multiplications systematically helps in managing larger problems efficiently. This strategy becomes increasingly useful as the number of integers increases.

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Most popular questions from this chapter

DISTRIBUTIVE PROPERTY Use the distributive property to rewrite the expression without parentheses. $$ -2 x(x-8) $$

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Write the verbal sentence as an equation or an inequality. Nine more than three fourths of \(y\) is less than six times \(w\)n.

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