Question

For Exercises \(79-86,\) explain the mistake in each problem and work it correctly. $$\begin{array}{l} -2[16(3-5)+7] \\ =-2[16(-2)+7] \\ =-2[32+7] \\ =-2[39] \\ =-78 \end{array}$$

Short answer

Answer: The correct solution of the given expression is 50.

Step-by-step solution

Observe the problem

First, let's read through the problem and identify its components. $$-2[16(3-5)+7]$$ We have a product of -2 and the expression \((16(3-5)+7)\). First, we need to find the error in the given solution.

Find the mistake

By examining the given steps, we can notice that the mistake occurs at the second step: $$-2[16(-2)+7] = -2[32+7]$$ Here, the issue is the handling of the multiplication of -2 and \(16(-2)\). It should have been \((16 * -2)\) instead of \(-2[32]\).

Correctly solve the problem

Now let's work through the problem correctly: 1. Evaluate the expression inside the parentheses: \((3-5) = -2\) $$-2[16(-2)+7]$$ 2. Apply the multiplication inside the brackets: \(16(-2) = -32\) $$-2[-32+7]$$ 3. Evaluate the expression inside the brackets: \(-32 + 7 = -25\) $$-2[-25]$$ 4. Multiply \(-2\) by \(-25\) to get the final result: $$-2(-25) = 50$$ The correct solution is \(50\).

Key concepts

Order of Operations

When solving algebraic expressions, it's essential to follow the correct sequence known as the order of operations. This mathematical convention determines which procedures to perform first to evaluate expressions accurately. The mnemonic 'PEMDAS' helps us remember the sequence: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

In the exercise \( -2[16(3-5)+7] \), the first step is to handle the parentheses (3-5), which correctly simplifies to -2, showcasing the 'P' in 'PEMDAS'. However, the mistake arises in the next step where the multiplication, the 'M' in 'PEMDAS', is handled incorrectly. This is a great pointer for students to always revisit PEMDAS whenever evaluating complex expressions.

Integer Multiplication

Understanding integer multiplication is crucial when evaluating algebraic expressions. When two integers with different signs are multiplied, the result is a negative number, whereas the product of two integers with the same sign is positive.

In our example, the multiplication of 16 and -2 wasn't done correctly in the presented steps. Remembering that a positive times a negative yields a negative is key. Hence, \( 16 \times -2 \) results in \( -32 \) and not \( 32 \), which can mislead to an incorrect final answer.

Evaluating Expressions

Evaluating expressions involves simplifying algebraic statements using the order of operations. Always start inside the innermost parentheses or brackets and work outwards, ensuring you follow the correct operation order.

The initial expression \( -2[16(3-5)+7] \) should first address the equation within the parentheses, as performed correctly in the example. However, the next critical step is to correctly evaluate the products and sums within the brackets before multiplying by -2. This step is often where errors may surface, so diligence and double-checking each operation are advised.

Algebraic Problem-Solving

Algebraic problem-solving requires a step-by-step approach to simplify and solve expressions. Identifying and correcting errors is a significant part of the learning process. Once you notice a mistake, work through the correct steps methodically, ensuring each operation complies with the order of operations.

Re-evaluating the original expression after each step can help prevent errors. In the exercise, the mistake was in misapplying integer multiplication and not adhering to the proper order of operations. To become proficient in algebraic problem-solving, practice these steps regularly with a variety of problems, and always review your calculations for potential errors.