Problem 52 - Algebra for College Students

Question

Simplify each of the numerical expressions. $$4(0.1)^{2}-6(0.1)+0.7$$

Step-by-Step Solution

Answer

The simplified expression is 0.14.

Step 1: Evaluate the Exponentiation

First, calculate the square of 0.1. Use the formula for exponentiation: \[ (0.1)^2 = 0.1 \times 0.1 = 0.01 \]

Step 2: Multiply with the Coefficient

Next, multiply the result by 4, as indicated in the expression: \[ 4 \times 0.01 = 0.04 \]

Step 3: Calculate the Second Term

Now multiply 6 by 0.1 from the term \(-6(0.1)\):\[ -6 \times 0.1 = -0.6 \]

Step 4: Combine Results

Finally, sum all the calculated terms to simplify the expression. Subtract and add the numbers sequentially:\[ 0.04 - 0.6 + 0.7 = 0.14 \]

Key Concepts

Exponentiation

Exponentiation is the process of raising a number, called the base, to the power of an exponent. In simple terms, it means multiplying the base by itself a certain number of times as indicated by the exponent. For instance, in the expression \((0.1)^2\), the base is 0.1 and the exponent is 2. This means we multiply 0.1 by itself once:

• \((0.1)^2\) translates to \(0.1 \times 0.1\).
• The result of \(0.1 \times 0.1\) is 0.01.

Exponentiation is a critical step in mathematical operations and often comes first in solving expressions to ensure accuracy in subsequent arithmetic operations.

Arithmetic Operations

Arithmetic operations include basic mathematical processes such as addition, subtraction, multiplication, and division. In dealing with expressions that involve arithmetic, it's crucial to perform each operation correctly. Looking at exercises like the one provided:

• Once the exponentiation is completed, multiply the result by the coefficient, which is often the number in front of a term. For example, \(4 \times 0.01 = 0.04\).
• Similarly, for subtraction, consider the term \(-6 \times 0.1\). The product is \(-0.6\).

These basic arithmetic operations are essential and must be performed in step-by-step order to achieve the right result.

Order of Operations

The order of operations is a set of rules that determines the sequence in which the operations are executed in a mathematical expression. Remember the acronym PEMDAS as a quick guide:

• P: Parentheses first
• E: Exponents (i.e., powers and square roots, etc.)
• MD: Multiplication and Division (from left to right)
• AS: Addition and Subtraction (from left to right)

In the problem provided, exponentiation is done first, followed by multiplication, and finally, the expression is simplified using addition and subtraction:

• First, solve \((0.1)^2 = 0.01\).
• Next, compute \(4 \times 0.01 = 0.04\) and \(-6 \times 0.1 = -0.6\).
• Lastly, sum the calculated values: \(0.04 - 0.6 + 0.7 = 0.14\).

Following the order of operations ensures that each part of the expression is simplified systematically, leading to the correct final result.