Problem 52 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

0.14

Step 1: Square the Number Inside the Parentheses

The first operation is to square the number inside the parentheses: \((0.1)^2 = 0.01\). This expression now looks like this: \[4(0.01) - 6(0.1) + 0.7\]

Step 2: Multiply Each Term by Its Coefficient

Multiply the squared term by 4: \[4 \times 0.01 = 0.04\]Then multiply the second term by its coefficient:\[6 \times 0.1 = 0.6\]Now the expression is:\[0.04 - 0.6 + 0.7\]

Step 3: Perform Subtraction and Addition

First, subtract 0.6 from 0.04 to get:\[0.04 - 0.6 = -0.56\]Then, add 0.7 to -0.56 to find the simplified result:\[-0.56 + 0.7 = 0.14\]

Key Concepts

Simplification

Simplifying a numerical expression means breaking it down into its simplest form. This process often involves removing unnecessary elements by combining like terms and performing arithmetic operations. To start, it's vital to understand and execute all operations following the order of operations, commonly known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). By focusing on simplifying, you effectively map out a clearer and more understandable expression.

To illustrate, in our example, we began by dealing with exponents within parentheses. We then moved systematically through each mathematical operation to transform the expression. Alongside consistent practice, improved simplification skills lead to enhanced clarity in solving complex expressions.

Multiplication

Multiplication is the process of adding a number to itself a certain number of times, and it plays a crucial role in simplifying expressions. In our exercise, we directly applied multiplication after simplifying the exponential part of the expression.

When multiplying a number by a coefficient, you must ensure accuracy as it significantly impacts the rest of your expression simplification. In our example, after calculating \(0.1^2\) as \(0.01\), we multiplied by 4 to get \(0.04\). Similarly, multiplying 6 by \(0.1\) resulted in \(0.6\).

Key points to remember during multiplication include:

• Check and double-check your multiplication to avoid mistakes.
• Pay attention to signs (positive or negative), as they determine the direction of your numerical outcome.
• Practice with different problems to reinforce your understanding.

By gaining mastery over multiplication, solving expressions simplifies easily and efficiently.

Addition and Subtraction

Addition and Subtraction are fundamental arithmetic operations that come into play after multiplication. In the simplification process, they often come last in the order of operations unless specific parentheses indicate otherwise. In our exercise, these operations were used to combine all the simplified components into a single result.

To simplify expressions effectively, it's vital to:

• First, complete any existing subtraction, like subtracting \(0.6\) from \(0.04\), giving \(-0.56\).
• Then, add any remaining terms, such as adding \(0.7\) to \(-0.56\), resulting in \(0.14\).
• Keep calculations organized to prevent confusing positive and negative outputs.

Remember that each arithmetic operation brings the expression closer to its most streamlined form. Building confidence in these arithmetic skills removes hurdles and helps manage complex calculations with ease.