Problem 51 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the order of operations to simplify each expression. $$\left[7+3\left(2^{3}-1\right)\right] \div 21$$

Step-by-Step Solution

Answer

The simplified solution of the given expression is \(8\).

Step 1: Solving within Parentheses

From the given expression \[ \left[7+3\left(2^{3}-1\right)\right] \div 21 \], let's first solve the operation inside parentheses \(\left(2^{3}-1\right)\). This is because according to the rules of 'Order of Operations', operations in parentheses have the highest precedence.

Step 2: Solving Exponents

Next, according to the 'Order of Operations', comes the evaluation of exponents. Here \[2^{3}\] equals \(8\). By substituting \[2^{3}\] with \(8\) the expression within Parentheses/Brackets becomes \[8 - 1 = 7\]. Now, replace \(\left(2^{3}-1\right)\) with \(7\) in the main expression.

Step 3: Performing Multiplication

Looking at the updated expression \[7 + 3 * 7 \div 21\], multiplication gets precedence before addition. So multiply \(3\) and \(7\) to get \(21\). Replace \(3 * 7\) with \(21\) in the equation, which then becomes \[ 7 + 21 \div 21 \].

Step 4: Performing Division and Addition

In the expression \[7 + 21 \div 21\], perform the division first which gives \(1\). Replace the \[21 \div 21\] with \(1\) then the expression becomes \[7 + 1\]. Finally, perform the addition operation which gives \(8\) as answer.

Key Concepts

Exponents

Exponents are a key component in math that indicate how many times a number, known as the base, is multiplied by itself. For example, the exponent in the expression \(2^3\) means 2 must be multiplied by itself 3 times: \(2 \times 2 \times 2\). This equals 8. Whenever you encounter exponents in an expression, compute them as the initial step after resolving everything inside parentheses. This is crucial because exponents simplify expressions quickly and ensure accurate calculations later on.
When simplifying expressions, always tackle exponents directly after dealing with any operations inside parentheses. By adhering to this order, your calculations remain consistent with the mathematical rules known as the order of operations.

Parentheses

Parentheses are used in mathematical expressions to group terms and operations that need to be solved first. In expressions like \([7 + 3(2^3 - 1)]\), parentheses organize the order and priority of operations. This ensures that calculations perform in a logically structured manner.
When tackling an expression, first focus on evaluating anything within parentheses. This might include separate calculations, such as subtraction or addition inside the parentheses. In the given problem, we first solved \(2^3 - 1 = 8 - 1 = 7\), significantly simplifying the expression by focusing inside the confines of the parentheses.

• Always look inside parentheses first.
• Handle any exponents, multiplication, or addition and subtraction within them before tackling outside operations.

By doing so, the sequence remains clear and results in the correct outcome.

Multiplication

Multiplication often appears in expressions alongside addition and subtraction. Understanding its position in the hierarchy of operations is crucial. For instance, multiplication comes directly after resolving any operations inside parentheses and calculating exponents.
In the expression \(7 + 3 \times 7\), multiplication takes priority over addition. This rule ensures that we compute \(3 \times 7 = 21\) before adding 7. Failing to respect this order would lead to incorrect results. Multiplication is an arithmetic operation used to simplify the numerical value of terms within an expression:

• When part of an expression, always perform multiplication before addition or subtraction, unless parentheses dictate otherwise.
• Multiplication simplifies terms by combining like pieces.

Always confirm that you've resolved any multiplication early in the process to keep subsequent steps straightforward.

Addition and Subtraction in Sequence

Addition and subtraction come at a later stage in the order of operations, but it's vital to manage them properly. They are processed sequentially from left to right, which maintains the structural integrity of the expression.
In the final stages of solving our expression, \(7 + 1\) follows the necessary operations where division was completed before adding. This reduces potential errors and ensures consistency in calculations.

• Once all higher-priority operations like multiplication and division are complete, address addition and subtraction.
• Move from left to right to keep operations orderly and accurate.

Finishing with addition and subtraction guarantees that the expression has been thoroughly and correctly evaluated.