Problem 1 - A Brief Introduction to Engineering Computation with MATLAB

Question

Determine the value of each of the following. $$ 6 \times 7+4^{2}-2^{4} $$

Step-by-Step Solution

Answer

The value of the expression \(6 \times 7+4^{2}-2^{4}\) is 42.

Step 1: Identify and simplify the terms with exponents

In our given expression, there are two terms with exponents: \( 4^{2} \) and \( 2^{4} \) Let's simplify each term: \( 4^{2} = 16 \) \( 2^{4} = 16 \) Now our expression becomes: $$ 6 \times 7 + 16 - 16 $$

Step 2: Perform the multiplication operation

Now, we'll perform the multiplication operation which is \( 6 \times 7 \). \( 6 \times 7 = 42 \) Our expression can now be simplified to: $$ 42 + 16 - 16 $$

Step 3: Perform the addition and subtraction operations from left to right

Finally, we will perform the addition and subtraction operations as they appear from left to right in the expression. First, we add: $$ 42 + 16 = 58 $$ Then, we subtract: $$ 58 - 16 = 42 $$ So the final value of the given expression is 42.

Key Concepts

Order of operations

When solving mathematical problems, it's crucial to follow the order of operations. This principle ensures you arrive at the correct result. The commonly used acronym for order of operations is PEMDAS, which stands for:

• P: Parentheses
• E: Exponents
• M/D: Multiplication and Division (from left to right)
• A/S: Addition and Subtraction (from left to right)

In our example, the expression to solve is \( 6 \times 7 + 4^2 - 2^4 \). Here, no parentheses are present, but we do have exponentiation, multiplication, and addition and subtraction. By sticking to the order, you'll manage each operation in the right sequence for a precise outcome.

Exponentiation

Exponentiation is a critical step in our mathematical expression. It involves raising a number (the base) to the power of another number (the exponent). In simple terms, it's multiplying the base by itself a certain number of times. Consider our expression: we have two powers, \( 4^2 \) and \( 2^4 \).

• To calculate \( 4^2 \), multiply 4 by itself, giving us \( 16 \).
• For \( 2^4 \), multiply 2 by itself 4 times, resulting in \( 16 \) again.

After handling the exponents first, our expression simplifies to \( 6 \times 7 + 16 - 16 \). It's important to always handle exponentiation before moving on to multiplication, addition, or subtraction to maintain accuracy.

Multiplication

Multiplication is a fundamental operation where we combine equal groups of numbers. It's denoted by the symbol "\(\times\)" and is one of the earlier steps, according to PEMDAS, as long as there are no parentheses or exponents to solve first.
For the expression \( 6 \times 7 + 16 - 16 \), after evaluating exponents, the next step is multiplication. Calculate \( 6 \times 7 \), which equals 42.
After completing this multiplication step, your expression simplifies further to \( 42 + 16 - 16 \). This precise execution of multiplication is vital in ensuring the solution follows the proper sequence.

Addition and Subtraction

Addition and subtraction are the final steps when evaluating an expression, executed from left to right as they appear. They're both equally important and should be completed after parentheses, exponents, and multiplication/division.
In our streamlined expression \( 42 + 16 - 16 \), we begin with addition: 42 plus 16 gives us 58.

• First, perform \( 42 + 16 \) to get 58.
• Then, handle the subtraction: \( 58 - 16 \), which results in 42.

Completing these steps successively ensures an accurate and reliable result, leading to the final solution. Practicing these operations deeply helps reinforce understanding and mastery of these basic arithmetic rules.