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Chapter 1: Problem 78

Simplify each expression. $$(5)\left(-\frac{2}{5}\right)$$

Short Answer

Expert verified
The simplified result of the expression \((5)\left(-\frac{2}{5}\right)\) is -2.

Step by step solution

01

Identify the terms

In the given expression, the terms to be multiplied are the integer 5 and the fraction \(-\frac{2}{5}\). Multiplication is the operation to be carried out.
02

Multiply the terms

To multiply the terms, simply multiply the integer with the numerator of the fraction and put it over the denominator. This results in: \(5 \times -\frac{2}{5}= -\frac{10}{5}\)
03

Simplifying the Fraction

-\frac{10}{5} simplifies to -2. This is because 10 divided by 5 results in 2.

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Key Concepts

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

Multiplying Fractions
Multiplying fractions may seem daunting at first, but with practice, it becomes a straightforward operation. When you multiply a fraction by another fraction, you simply multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. For example, \( \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \). However, when multiplying a fraction by an integer, think of the integer as a fraction with a denominator of 1. This way, you multiply the integer by the numerator of the fraction and keep the original fraction's denominator. It's often a good practice to simplify your result, if possible, to make the answer more understandable.
Simplifying Fractions
Simplifying fractions is an essential skill. To simplify a fraction, find the greatest common divisor (GCD) of the numerator and the denominator, then divide both by that number. For example, \( \frac{10}{5} \) simplifies to \( \frac{2}{1} \) or just 2, since the GCD of 10 and 5 is 5. Simplification helps make fractions easier to work with and, in many cases, may be required to reach the final answer in its most reduced form.
Algebraic Operations
Algebraic operations are the building blocks of algebra and include addition, subtraction, multiplication, and division. These operations can be used with numbers, variables, and expressions. In the context of simplifying expressions like \( 5 \left(-\frac{2}{5}\right) \), multiplication is the key operation. It's essential to remember the order of operations (PEMDAS鈥擯arentheses, Exponents, Multiplication and Division, Addition and Subtraction) to simplify expressions correctly. Algebraic operations also adhere to properties such as the distributive property, which states that \( a(b + c) = ab + ac \) and is often used when working with algebraic expressions.
Integer and Fraction Multiplication
When multiplying an integer by a fraction, consider the integer as a fraction with a denominator of 1. This means you can apply the same rules for multiplying fractions. For example, when multiplying 5 (which can be written as \( \frac{5}{1} \) ) and \( -\frac{2}{5} \), we multiply the numerators together, getting -10, and since the denominators are 1 and 5, we multiply those to get 5. The result \( -\frac{10}{5} \) can be simplified further to -2. It's a useful way to combine different kinds of numbers (integers and fractions) into a simplified algebraic expression.

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