Problem 52 - College Algebra and Calculus: An Applied Approach

Question

Use a calculator to evaluate the expression. (Round to two decimal places.)\(2\left(-7+\frac{1}{6}\right)\)

Step-by-Step Solution

Answer

-13.66

Step 1: Calculate the Inner Bracket

According to the order of operations, calculations within brackets should be carried out first. In this case, evaluate \(-7+\frac{1}{6}\). This results in -6.83, when rounded to two decimal places.

Step 2: Multiply Out the Result from the Bracket

Next step is to multiply the result from the previous step by 2. Evaluating \(2 \times (-6.83)\) gives -13.66.

Step 3: Round to Two Decimal Places

Lastly, ensure the result is rounded correctly to two decimal places. In this case, the result is already rounded to two decimal places, so the final result remains -13.66.

Key Concepts

Evaluating Expressions

When evaluating mathematical expressions, it's important to follow the order of operations known as PEMDAS or BIDMAS: Parentheses/Brackets, Exponents/Indices, Multiplication and Division, and finally, Addition and Subtraction. This specific order ensures accurate calculations. Let's break this down using the provided expression.

• First, look for parentheses or brackets in the expression. In the example, we need to calculate the expression inside the brackets \(-7 + \frac{1}{6}\).
• Solving \(-7 + \frac{1}{6}\) means performing the addition or subtraction as indicated. Note that \(-7\) is equal to \(-7.00\). So, the tiny fraction \(\frac{1}{6}\) affects the total slightly.
• Once evaluated, the expression within the parentheses simplifies to \-6.83\ (rounded to two decimal places).

By carefully following each operation in sequence, expressions can be simplified and evaluated accurately, leading to a clear solution.

Rounding Numbers

Rounding numbers is an essential skill in mathematics, especially when dealing with decimals. It helps simplify figures by reducing the number of digits. Here's how you can round numbers effectively.

• Identify which decimal place you need to round to. For example, rounding to two decimal places means leaving two digits after the decimal point.
• Look at the digit immediately after your desired decimal place. If it is 5 or greater, round up the last considered digit by one. If it's less than 5, keep the last considered digit as it is.
• In our expression \(-6.83\), it's already rounded to two decimal places. This is because the third decimal place is irrelevant to the two decimal place requirement.

Rounding not only aids in simplifying numbers but ensures calculations remain manageable and comprehensible.

Mathematical Calculations

Mathematical calculations often involve multiple operations and may require using tools like calculators for precision. Let's explore these concepts as reflected in the given exercise.

• A calculator is beneficial for handling complex arithmetic, especially for fractions and to maintain accuracy in calculations.
• When using a calculator, carefully input each part of the expression to avoid errors. For example, \(2 \times (-6.83)\) indicates the multiplication of two numbers: \(-6.83\) multiplied by 2.
• The result, \(-13.66\), shows the product of the previous step. Calculators ensure each operation is done precisely, minimizing human error.

By understanding each operation's role and using available tools like calculators, mathematical calculations become less daunting and more precise.