Problem 40 - Fundamentals of Mathematics

Question

Find the value of each expression. $$3[16-3(a+3 b)] \text { , if } a=3 \text { and } b=-2$$

Step-by-Step Solution

Answer

The value of the expression is 75.

Step 1: Substitute Values for Variables

The expression given is \(3[16-3(a+3b)]\). Substitute \(a = 3\) and \(b = -2\) into the expression. This gives: \[3[16 - 3(3 + 3(-2))].\]

Step 2: Simplify Inside the Parentheses

Within the expression \(3 + 3(-2)\), first calculate \(3(-2)\). This gives: \[3 - 6 = -3.\] So the expression now is \[3[16 - 3(-3)].\]

Step 3: Multiply Inside the Brackets

Next, calculate \(3(-3)\), which is \(-9\). Substitute back into the expression to obtain: \[3[16 - (-9)].\]

Step 4: Simplify the Expression Inside the Brackets

Apply the negative sign inside the brackets: \(16 - (-9) = 16 + 9 = 25\). So our expression simplifies to: \[3[25].\]

Step 5: Final Multiplication

Multiply the result by 3: \(3 \times 25 = 75\). So the final value of the expression is 75.

Key Concepts

Substituting Variables

Substituting variables is one of the foundational skills in algebra. When you're given an algebraic expression that includes variables, substitution simply means replacing these variables with the specific values given. It's like solving a puzzle by filling in the blanks with the correct pieces.
In our example, we start with the expression \(3[16 - 3(a+3b)]\). Here, \(a\) and \(b\) are the variables. The problem states \(a = 3\) and \(b = -2\).
To substitute these values:

• Replace every \(a\) in the expression with 3.
• Replace every \(b\) in the expression with \(-2\).

This results in modifying the original expression to \(3[16 - 3(3+3(-2))]\), making it clear and ready for the subsequent steps in the solving process.

Order of Operations

Handling expressions correctly relies a lot on the order of operations. These are specific rules that dictate the sequence in which operations should be performed to ensure consistent results. In mathematics, we often remember this order using the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

In our expression \(3[16 - 3(3 + 3(-2))]\), follow the order of operations:

• Begin by tackling what's inside the parentheses \((3 + 3(-2))\).
• Inside there, first compute what is in the innermost parentheses or directly alongside a negative number: \(3(-2)\) which simplifies to \(-6\).
• Then sum it with 3 to get \(-3\).

Now our expression is \(3[16 - 3(-3)]\). Continue executing operations inside the brackets first before resolving outside. This sequence ensures accuracy no matter how complicated expressions might become.

Simplifying Expressions

Simplifying expressions involves combining like terms and performing the calculations until a single, simplified result is reached. This step is crucial for finding the final answer to an algebraic expression.

With our modified expression of \(3[16 - 3(-3)]\), the next task is to simplify within the brackets. Calculate \(-3(-3)\), which yields 9.

1. Because you have a "double negative" here, it effectively becomes a positive operation: \(16 + 9\).
2. This further simplifies to \(25\).

Ultimately, we multiply by 3: \(3 \times 25\).
The steps of simplifying bit by bit make it easier to track operations and ensure clarity in reaching the final result, which in this example is 75. That way, you can be confident in the correctness of your answer.