Problem 10 - Technical Mathematics with Calculus

Question

Combined Operations with Exact Numbers. Perform each computation by calculator. $$(422+113-533)^{4}$$

Step-by-Step Solution

Answer

The result of the operation \( (422 + 113 - 533)^4 \) is 16.

Step 1: Perform the operations within the parenthesis

Start by adding and subtracting the numbers inside the parenthesis. Compute the sum of 422 and 113, then subtract 533 from the result.

Step 2: Calculate the sum and difference

First, calculate the sum of the first two numbers: 422 + 113 equals 535. Then subtract 533 from this sum: 535 - 533 equals 2.

Step 3: Raise the result to the fourth power

Now, take the result obtained from the operations within the parenthesis, which is 2, and raise it to the fourth power. Calculate 2 to the power of 4.

Step 4: Obtain the final result

Calculating 2 to the power of 4 gives us 2 * 2 * 2 * 2, which equals 16.

Key Concepts

Order of Operations

Understanding the order of operations, often encapsulated by the acronym PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction), is fundamental in executing combined arithmetic calculations correctly.

When solving a mathematical expression, this hierarchy guides us to compute operations in the correct sequence. Begin with operations inside parentheses, followed by exponents, then multiplication and division from left to right, and, finally, addition and subtraction from left to right.

In the exercise (422+113-533)^4, the parentheses indicate that we should perform the addition and subtraction of the numbers 422, 113, and -533 first. Failure to follow the order of operations could lead to incorrect results and confusion.

Exponentiation

Exponentiation is an arithmetic operation that involves raising a number, known as the base, to the power of an exponent. The exponent indicates how many times the base is multiplied by itself.

For instance, raising 2 to the fourth power is denoted as 2^4, which means multiplying 2 by itself four times: 2 x 2 x 2 x 2. In the case of our exercise, once we've calculated the operations inside the parentheses to get 2, we then raise it to the fourth power to find the solution.

Arithmetic Operations

Arithmetic operations include addition, subtraction, multiplication, and division. These basic operations are the building blocks of mathematics.

• Addition combines numbers to get a sum.
• Subtraction finds the difference between numbers.
• Multiplication is repeated addition, combining equal groups.
• Division splits a number into equal parts.

Using these operations with precision is crucial, especially when performing combined operations with exact numbers, as exemplified in our exercise with the expression (422+113-533)^4. Here, we apply addition and subtraction to find the number which we will then exponentiate.