Problem 66 - Elementary Algebra

Question

Using the result of problem 52 , is it correct to write \((a+b)=a+b ?\)

Step-by-Step Solution

Answer

Short Answer: Yes, the expression \((a+b)=a+b\) is correct. The parentheses do not affect the expression since there are no numbers or operations outside the parentheses, and both expressions represent the same operation (adding \(a\) and \(b\)).

Step 1: Understand the Expression

First, let's understand what the expression is asking. We have two variables, \(a\) and \(b\), and we are asked whether adding the variables with parentheses around them, \((a+b)\), is the same as just adding the variables without the parentheses, \(a+b\).

Step 2: Parentheses in Algebraic Expressions

Parentheses are generally used in algebraic expressions to group terms together and to show the correct order of operations. They are used to indicate that the operations inside the parentheses should be performed first before any other operations outside the parentheses. In the expression \((a+b)\), parentheses are used to group \(a\) and \(b\) together.

Step 3: Distributive Property

The Distributive Property states that for any real numbers \(a,b,c\), we have \(a(b+c) = ab + ac\). In other words, when we have a product of a number and a sum inside parentheses, we can rewrite the expression by multiplying each term in the parentheses by the number outside the parentheses.

Step 4: Applying the Distributive Property

In our case, there are no numbers outside the parentheses, so we do not have to apply the Distributive Property. \((a+b)\) can remain as it is, without any modification.

Step 5: Comparing Expressions

Now, we can compare the expressions \((a+b)\) and \(a+b\). We see that both expressions are the same because there are no numbers or operations outside the parentheses in \((a+b)\). The parentheses do not affect the expression, so they can be removed. Therefore, \((a+b)=a+b\) is a correct statement.

Key Concepts

Order of Operations

Understanding the order of operations is essential in algebra. It helps us determine which part of an expression to solve first. You might remember this order using the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This hierarchy ensures that everyone follows the same steps and gets the same result when solving an equation.
For example, if you see the expression \(3 + 5 \times 2\), you should multiply first and then add: \(5 \times 2 = 10\), then \(3 + 10 = 13\).
The order of operations ensures clarity and consistency in mathematical expressions, preventing confusion and errors.

Distributive Property

The distributive property is a useful tool in algebra when dealing with expressions involving multiplication and addition. This property tells us that for any numbers \(a, b,\) and \(c\), the equation \(a(b+c) = ab + ac\) holds true.
Basically, you can "distribute" the multiplication across the terms inside parentheses. Imagine you have \(2(3 + 4)\). According to the distributive property, you can turn it into \(2 \times 3 + 2 \times 4 = 6 + 8 = 14\).
This property becomes incredibly useful in simplifying complex expressions and solving equations where multiplication interacts with a sum or difference in parentheses.

Parentheses in Algebra

Parentheses play a crucial role in algebraic expressions. They indicate which operations should be performed first and can significantly change the outcome of an expression.
For instance, in the expression \((2 + 3) \times 4\), you need to add first due to the parentheses: \(2 + 3 = 5\), then multiply: \(5 \times 4 = 20\).

• Parentheses help in organizing terms, especially in complex equations.
• They can denote multiplication, as in \(3(x + 2)\), meaning multiply \(3\) by everything inside the parentheses.
• Understanding how parentheses affect calculations prevents mistakes and misunderstandings.

Parentheses ensure that the intended calculations are clear, highlighting the importance of grouping operations correctly.