Problem 31 - Fundamentals of Mathematics

Question

Use the order of operations to determine each value. \(64 \cdot\left(3^{2}-2^{3}\right)\)

Step-by-Step Solution

Answer

The value is 64.

Step 1: Evaluate Exponents

Begin by evaluating the exponents in the expression. Calculate each term with an exponent:- The exponent in the term \(3^2\) is 9 because \(3^2 = 3 \times 3 = 9\).- The exponent in the term \(2^3\) is 8 because \(2^3 = 2 \times 2 \times 2 = 8\).Now the expression is \(64 \cdot (9 - 8)\).

Step 2: Simplify Inside the Parentheses

Next, perform the subtraction inside the parentheses:- Subtraction inside the parentheses: \(9 - 8 = 1\).Now the expression simplifies to \(64 \cdot 1\).

Step 3: Perform Multiplication

Finally, multiply the result by 64:- Perform the multiplication: \(64 \times 1 = 64\).

Key Concepts

Exponents

Exponents are an essential concept in mathematics that indicate how many times a number, known as the base, is to be multiplied by itself. For instance, in the expression \(3^2\), \(3\) is the base, and \(2\) is the exponent, meaning you multiply 3 by itself once: \(3 \times 3 = 9\).

• Exponents can be thought of as a form of repeated multiplication.
• They simplify expressions and make calculations more manageable.

In the given exercise, you deal first with the exponents of both 3 and 2. Calculating these exponents is the initial step in applying the order of operations. Always calculate exponents before moving on to other operations like addition, subtraction, or multiplication.
By evaluating \(3^2\) and \(2^3\), you transform the expression into something simpler to work with, as seen in the original solution.

Parentheses

Parentheses are vital in mathematical expressions because they indicate which operations should be completed first. In expressions without parentheses, the standard order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (left-to-right), Addition and Subtraction (left-to-right)), guides how you solve the problem.
When parentheses are present, any operations within them are given top priority. In our example, once the exponents are dealt with, the result is placed inside the parentheses: \(9 - 8\). This subtraction must be completed before you move on to any operations outside of the parentheses.

• Always resolve operations inside parentheses first.
• This might include addition, subtraction, multiplication, or even another set of exponents.

Resolving what's inside the parentheses helps ensure that calculations are accurate and adhere to mathematical conventions.

Multiplication

Multiplication is repeated addition. In many mathematical problems, it comes after parentheses and exponents in the order of operations. After resolving the expression within the parentheses, as in the exercise, the next step focuses on multiplying any resultant value by the numbers outside, known in our case as \(64 \times 1\).
Multiplication is straightforward but crucial when dealing with longer expressions. Finding the product provides the final value of the expression.

• Ensure all previous operations inside exponents and parentheses are completed before performing multiplication.
• This is the last step in resolving expressions under PEMDAS when division is not involved.

In this specific example, solving \(64 \cdot 1\) yields a final solution of 64, demonstrating how multiplication ties together operations in an expression.