Problem 9 - Technical Mathematics with Calculus

Question

Combined Operations with Exact Numbers. Perform each computation by calculator. $$(5+6)^{2}$$

Step-by-Step Solution

Answer

The calculated result is 121.

Step 1: Evaluate the Addition Inside the Parentheses

Start by evaluating the expression inside the parentheses. In this case, you need to add the numbers 5 and 6 together.

Step 2: Apply the Exponentiation

After adding the numbers inside the parentheses, you will have a single number. The next step is to square this number, which means multiplying the number by itself.

Step 3: Final Step: Calculate the Result

Using a calculator, take the result from Step 1 and multiply it by itself to get the final answer.

Key Concepts

Calculator Computation

Performing calculations today has been greatly expedited thanks to the use of calculators. To solve an exercise like \( (5+6)^2 \) without error, it's critical to know how to use a calculator effectively. Firstly, ensure your calculator is in the correct mode for arithmetic computations (typically the standard mode). Enter the values and expressions exactly as they appear in the exercise.

For most calculators, you must enter the parenthesized addition first, followed by the exponentiation. You may also find a specific button for exponentiation often labeled as 'x^2' or similar representations for squaring a number. The sequence of buttons pressed should reflect the order of operations, a fundamental concept in executing mathematical computations accurately.

Order of Operations

Underlying calculator computation is the principle of 'order of operations'. It's a set of rules detailing the sequence in which parts of a mathematical expression should be solved. The acronym PEMDAS helps to remember this sequence:

• \(P\) for Parentheses: Solve expressions inside parentheses first.
• \(E\) for Exponents: After parentheses, calculate powers and roots.
• \(MD\) for Multiplication and Division: Work these out from left to right.
• \(AS\) for Addition and Subtraction: These are the final steps, also executed from left to right.

In our problem \( (5+6)^2 \), the parentheses take priority. After computing the sum, we then apply exponentiation. It is vital to follow these rules to avoid mistakes and reach the correct solution.

Exponentiation

Exponentiation is an arithmetic operation representing the repeated multiplication of a number by itself. It involves a base number and an exponent. For example, in the expression \( a^n \) , \(a\) is the base and \(n\) is the exponent, indicating how many times the base multiplies itself.

When squaring a number, as in our problem \( (5+6)^2 \), the base \( (5+6) \) is multiplied by itself only once because the exponent \(2\) denotes two factors of the base. It is crucial to understand that the exponent affects only the number it directly follows, so only the result inside the parentheses is squared, not each individual number (5 or 6) inside the parentheses.