Problem 49 - Algebra for College Students

Question

Simplify each of the numerical expressions. $$2^{4}-2(2)^{3}-3(2)^{2}+7(2)-10$$

Step-by-Step Solution

Answer

The simplified expression is -8.

Step 1: Understanding the Problem

We are given the mathematical expression \(2^{4} - 2(2)^{3} - 3(2)^{2} + 7(2) - 10\). Our task is to simplify this expression by carefully following the order of operations (PEMDAS/BODMAS).

Step 1: Calculate Exponents

In the expression, identify and calculate all exponents. We calculate \(2^{4}\), \(2^{3}\), and \(2^{2}\).\(2^{4} = 16\)\(2^{3} = 8\)\(2^{2} = 4\)

Step 2: Multiply and Combine Terms

Substitute the values from the previous step back into the expression and perform necessary multiplications:\(16 - 2 \times 8 - 3 \times 4 + 7 \times 2 - 10\).Calculate the multiplications:\(2 \times 8 = 16\)\(3 \times 4 = 12\)\(7 \times 2 = 14\)

Step 3: Simplify the Expression

Substitute the multiplication results into the expression:\(16 - 16 - 12 + 14 - 10\).Now, perform the additions and subtractions from left to right:First, \(16 - 16 = 0\).Next, \(0 - 12 = -12\).Then, \(-12 + 14 = 2\).Finally, \(2 - 10 = -8\).

Step 5: Conclusion

Simplifying the expression gives us the final result: \(-8\).

Key Concepts

Order of Operations

When simplifying numerical expressions, the Order of Operations is crucial to get the correct answer. Often remembered by the acronym PEMDAS or BODMAS, which stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction, this rule dictates the sequence in which operations should be carried out. This is essential because different orders can lead to different results.

• Parentheses: Handle any operations inside parentheses first, although not present in this specific problem, they're fundamental when they appear.
• Exponents: Next, evaluate all exponential expressions available in the equation.
• Multiplication and Division: Then, perform these from left to right, as they appear in the expression.
• Addition and Subtraction: Finally, complete any addition and subtraction from left to right.

Understanding and applying this sequence reliably ensures that complex expressions are simplified accurately. It's a step-by-step approach that reduces errors and confusion.

Exponents

Exponents are a way of expressing repeated multiplication of the same number. In our problem, the base of the exponents is 2, highlighted in terms like \(2^4\), \(2^3\), and \(2^2\). When you see a term with an exponent, it tells you to multiply the base number by itself a certain number of times.

• \(2^4\) ("two to the fourth power"): This means 2 multiplied by itself four times which equals 16.
• \(2^3\): Here, 2 is multiplied by itself three times, resulting in 8.
• \(2^2\): This is 2 times 2, giving 4.

By calculating these exponent values first, as per the order of operations, you can then substitute them back into the expression for further simplification.

Multiplication and Division

Following exponents in the order of operations comes multiplication and division. In any expression, these operations are performed from left to right.

In this exercise, once we calculated the exponents, each was part of a multiplication:

• The term \(2 \times 8\) where 8 was the result of \(2^3\), this equals 16.
• The term \(3 \times 4\), with 4 being the result of \(2^2\), ends up being 12.
• Finally, \(7 \times 2\) equals 14, straightforward multiplication.

By substituting these products back into the expression, we consolidate terms for easier subtraction and addition in the next steps. Thoroughly mastering multiplication and division ensures that you're stepping smoothly towards the correct final result.