Problem 49 - College Algebra

Question

Exercises will help you prepare for the material covered in the next section. In each exercise, perform the indicated operation or operations. $$ \frac{1}{2}[8-(-8)] $$

Step-by-Step Solution

Answer

The result of the operation \( \frac{1}{2}[8 + 8] \) is 8.

Step 1: Resolve Inside the Brackets

The operation inside the brackets is 8 - (-8). In mathematics, subtracting a negative number is equivalent to adding a positive number. Hence, this operation becomes \(8 + 8 = 16\).

Step 2: Multiply by the Fraction

Multiply the result obtained from the brackets by the fraction 1/2. So, \(0.5 * 16 = 8\).

Key Concepts

addition and subtraction

Addition and subtraction are fundamental arithmetic operations that involve combining or taking away quantities. When performing operations with negative numbers, keep in mind a few essential points:

• Adding a positive number equivalent to increasing the value.
• Adding a negative number is just like subtracting its absolute value.
• Subtracting a negative number is the same as adding its opposite (positive value).

For example, in the exercise "8 - (-8)", subtracting \(-8\) is the same as adding \(+8\). Thus, the expression becomes \(8 + 8 = 16\). Remember this rule: two negatives make a positive when directly linked by a subtraction operation.

order of operations

The order of operations, often remembered by the acronym PEMDAS, outlines the sequence to correctly solve mathematical expressions:

• Parentheses - Solve operations inside them first.
• Exponents - Address powers and roots next.
• Multiplication and Division - Process from left to right.
• Addition and Subtraction - Also processed from left to right.

Brackets or parentheses are prioritized in mathematical calculations, ensuring clarity and accuracy. In our original exercise, we notice brackets around "8 - (-8)", so we solve this portion first before anything else. Only after resolving the parentheses can we perform any multiplication such as, in our case, multiplying with \(\frac{1}{2}\). Always keep PEMDAS in mind when approaching complex equations.

fraction multiplication

Fraction multiplication is crucial in solving many math problems and is, in fact, simpler than it seems. When multiplying by a fraction, you essentially scale down or up depending on whether your fraction is less than or more than 1.To multiply fractions:

• Multiply the numerators (top numbers) together.
• Multiply the denominators (bottom numbers) together.

In this exercise, we deal with multiplying the number 16 by \(\frac{1}{2}\). Here’s how you do it:

• Visualize 16 as \(\frac{16}{1}\).
• Multiply the numerators: \(16 \times 1 = 16\).
• Multiply the denominators: \(1 \times 2 = 2\).
• Simplify to get \(\frac{16}{2} = 8\).

Multiplying by \(\frac{1}{2}\) effectively finds half of the original number, aligning perfectly with our result.