Problem 19 - Prealgebra and Introductory Algebra

Question

Simplify. $$(4.1-3.9)-0.7^{2}$$

Step-by-Step Solution

Answer

The simplification of the given expression \((4.1 - 3.9) - 0.7^2\) results in \(-0.29\).

Step 1: Perform the first subtraction

First, simplify the expression inside the parenthesis. i.e., \(4.1 - 3.9\). This results in \(0.2\).

Step 2: Raise the decimal to power 2

Next, raise \(0.7\) to the power of 2 (i.e., square \(0.7\)), which gives \(0.49\).

Step 3: Perform the final subtraction

Finally, subtract the result of Step 2 from the result of Step 1. i.e., \(0.2 - 0.49 = -0.29\).

Key Concepts

Subtraction

Subtraction is one of the basic arithmetic operations, which involves finding the difference between two numbers or quantities. In this exercise, subtraction was used twice. First, within the parenthesis, we have the subtraction of decimals: 4.1 and 3.9.

To simplify an expression like this:

• Align the decimal points vertically. This helps ensure accuracy when dealing with decimal numbers.
• Subtract the numbers starting from the rightmost digits (tenths place in this example) towards the left.
• Calculate the difference. For instance, \(4.1 - 3.9 = 0.2\), because when you have 4.1 items and remove 3.9, you are left with 0.2 items.

Once you understand how to subtract decimals, it becomes easier to solve more complex expressions.

Exponents

Exponents represent repeated multiplication of a number by itself. When you see a number followed by an exponent, it signifies how many times the number is being multiplied. In this exercise, you encounter the expression:

• \(0.7^{2}\).

This means you multiply 0.7 twice by itself:

\[0.7 \times 0.7 = 0.49\]

The "2" here is the exponent, indicating that we square 0.7. It's essential to remember:

• An exponent of "2" signifies squaring; "3" would mean cubing the number, and so on.
• Always apply the exponent to the number itself, regardless if it's a whole number or a decimal, as shown above.

Decimal Operations

Operations with decimals function similarly to those with whole numbers, but require careful handling of the decimal point to ensure precision and accuracy. Decimal operations include addition, subtraction, multiplication, and division. This exercise specifically utilizes subtraction and multiplication (through exponentiation) with decimals.

Here are key points to remember:

• Always align decimals vertically when performing addition and subtraction to maintain accuracy.
• When multiplying decimals, such as in exponentiation, count the total number of decimal places from the numbers being multiplied.
  Then, the result will have that sum of decimal places. For example, \(0.7 \times 0.7\) involves 1 decimal place in each number, so the result, 0.49, has 2 decimal places.
• Subtraction of decimals follows after these operations; always ensure your calculation is double-checked.