/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 65 \(6(3+13)=6 \cdot 3+6 \cdot 13\)... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 65

\(6(3+13)=6 \cdot 3+6 \cdot 13\)

Short Answer

Expert verified
Yes, the exercise is correct, since '6(3+13)=6 \cdot 3+6 \cdot 13' simplifies to '96=96', which demonstrates the distributive property. So, the left side is equal to the right side when the distributive property is applied.

Step by step solution

01

Expression Setup

First observe the given equation, which is \(6(3+13)=6 \cdot 3+6 \cdot 13\). It's clear that the exercise wants you to prove the distributive property.
02

Application of Distributive Property

In order to apply the distributive property, you take the factor which is outside the parentheses (in this case, 6) and multiply it by each of the terms within the parentheses (in this case, 3 and 13). It gives us: \(6 \cdot 3 + 6 \cdot 13\).
03

Simplification of Expression

The next step is to simplify the expression. Multiply 6 by 3 to get 18, and multiply 6 by 13 to get 78. The simplified expression is \(18 + 78\).
04

Final Evaluation

Finally, add 18 and 78 together to get the final answer, which is 96.

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Key Concepts

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

Algebra Concepts
Algebra is a fundamental part of mathematics that deals with variables, equations, and formulas. It helps us understand mathematical relationships and solve problems using symbols. The distributive property is a critical algebraic concept used to simplify equations involving multiplication over addition or subtraction.
  • The distributive property states that multiplying a number by a sum is the same as multiplying the number by each addend separately and then adding the products. In mathematical terms, this property is expressed as: \( a(b + c) = ab + ac \).
  • The distributive property helps simplify and solve more complicated expressions by breaking them down into smaller, more manageable pieces.
In our original exercise, the distributive property is applied directly. By distributing the 6 over the expression \( (3 + 13) \), we simplify the equation step-by-step. This highlights a core principle in algebra: simplifying expressions through properties like distribution is essential for efficient problem-solving.
Multiplication
Multiplication is among the basic arithmetic operations and is crucial in algebra for simplifying expressions and solving equations. When applying multiplication in algebra, especially using the distributive property, you manage the process by multiplying terms systematically.
  • In our exercise, 6 is multiplied by each term within the parentheses: 3 and 13. This results in two separate products (\(6 \cdot 3\) and \(6 \cdot 13\)).
  • Multiplication in the scope of the distributive property allows us to distribute one factor across each term in an expression enclosed by parentheses.
  • This operation often simplifies complex expressions, making them easier to handle and solve.
By breaking down multiplication into these smaller tasks, you maintain control over the process and can easily arrive at the solution. Understanding multiplication as both a basic arithmetic operation and a tool in algebra is essential for mastering more advanced mathematical concepts.
Order of Operations
The order of operations is a critical rule in mathematics that specifies the correct sequence to follow when performing calculations. It ensures that everyone solves equations consistently and accurately. The common acronym PEMDAS helps remember this order: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
  • In our exercise, the distributive property aligns with the order of operations. We start with the expression inside the parentheses, \((3 + 13)\), but don't perform addition first because distribution takes precedence here.
  • Applying the distributive property effectively prioritizes multiplication over addition due to its role in simplifying expressions by breaking them down.
  • Once the multiplication is done separately, the results are added (\(18 + 78\) = 96).
Understanding the order of operations is vital, as skipping or rearranging the steps can lead to incorrect results. By adhering to these rules, you ensure that calculations are done correctly and efficiently, leading to the right solution.

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Most popular questions from this chapter

\(18\left(\frac{1}{2}+\frac{2}{3}\right)\)

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