Problem 17 - Prealgebra and Introductory Algebra

Question

Simplify. $$(1.2)^{2}-4.1(0.3)$$

Step-by-Step Solution

Answer

The answer is 0.21.

Step 1: Resolve Exponent

Begin by resolving the exponent. Calculate the square of 1.2: \( (1.2)^{2} \) evaluates to 1.44.

Step 2: Resolve Multiplication

Next, perform the multiplication operation. Multiply 4.1 by 0.3: \( 4.1 * 0.3 = 1.23 \).

Step 3: Perform Subtraction

Finally, subtract the two results obtained from the previous steps: 1.44 - 1.23 gives us the simplified resultant answer.

Key Concepts

Understanding Exponents

Exponents are a way to express repeated multiplication of a number by itself. When you see something like \((1.2)^2\), it means you multiply 1.2 by itself:

\(1.2 \times 1.2 = 1.44\).

Key points to remember about exponents include:

• The base is the number being multiplied, in this case, 1.2.
• The exponent tells you how many times to multiply the base by itself.
• \(a^2\) is often read as "a squared," where "a" is the base.

Exponents simplify writing and calculating larger numbers and are fundamental in algebra. With practice, you'll get comfortable recognizing and calculating them.

Mastering Multiplication

Multiplication is the process of combining equal groups. In the expression \(4.1 \times 0.3\), you’re scaling 4.1 by a factor of 0.3.

Here's how to think about multiplication:

• Consider multiplication as finding the total of several groups of things. For instance, 4.1 groups of 0.3 each.
• It’s always commutative, meaning \(a \times b = b \times a\).

In our case, multiplying 4.1 by 0.3 gives:

\(4.1 \times 0.3 = 1.23\).

Understanding how multiplication scales numbers, even with decimals, is key to working through more complex algebraic problems.

Simplifying with Subtraction

Subtraction is the process of taking away one quantity from another. In the exercise, you subtract the product from the exponent result:

\(1.44 - 1.23\).

Here's how subtraction fits into solving expressions:

• Start with the total and decrease it by the number you're subtracting.
• It’s used to find the difference between numbers.

In this problem, it simplifies to:

\(1.44 - 1.23 = 0.21\).

This final result represents the simplified expression. When dealing with subtraction, remember to align your decimals properly to avoid mistakes.