Problem 79 - Prealgebra

Question

Simplify the given expression. \((-8.1)(9.4)-1.8^{2}\)

Step-by-Step Solution

Answer

The simplified expression is \\(-79.38\\).

Step 1: Simplify Multiplication

The given expression is \((-8.1)(9.4) - 1.8^2\). First, let's simplify the multiplication \((-8.1)(9.4)\). Multiply \-8.1\ by \9.4\ to get the result. Calculating gives: \-76.14\.

Step 2: Evaluate the Exponent

Now, evaluate the square of the number \(1.8\), which is the \(1.8^2\) part of the expression.Calculating \(1.8 imes 1.8\) gives \3.24\.

Step 3: Substitute Values back into the Main Expression

Substitute the values from the previous steps back into the main expression: \(-76.14 - 3.24\).

Step 4: Perform Subtraction

Perform the subtraction in the expression: \(-76.14 - 3.24\). Combine the numbers by subtracting \3.24\ from \-76.14\ resulting in \-79.38\.

Key Concepts

Simplifying Expressions

Simplifying expressions is like tidying up a messy room; it makes complex mathematical expressions more manageable and easier to understand. In prealgebra, simplifying involves breaking down equations to their simplest form. This usually means:

• Carrying out operations like multiplication and division.
• Handling any exponents involved.
• Combining like terms by addition or subtraction.

In our exercise, we simplify the expression \((-8.1)(9.4) - 1.8^{2}\).First, by performing the multiplication, then evaluating the exponent, and finally by subtracting the results. This step-by-step approach ensures that each part of the expression is correctly simplified before combining all parts into the final answer.

Multiplication

Multiplication is one of the basic arithmetic operations that results in combining groups of equal quantities. When you multiply two numbers,

• One acts as the number of groups (the multiplier).
• The other is the size of each group (the multiplicand).

In our expression, \(-8.1 imes 9.4\), the operation means multiplying the negative number \(-8.1\) by \(9.4\).Negative numbers require special consideration; multiplying two numbers with differing signs results in a negative product. Thus, the multiplication goes as follows:\((-8.1)\times(9.4) = -76.14\).The negative sign in the product is crucial and ensures an accurate simplification later on.

Exponents

Exponents indicate how many times a number (the base) is used as a factor in a multiplication. The exponent is shown as a small number above and to the right of the base. In the expression\(1.8^2\),the number \(1.8\) is the base, and the exponent \(2\) tells us to multiply \(1.8\) by itself once:\(1.8 \times 1.8 = 3.24\).Exponents are part of what we call 'orders of operations' in mathematics, and usually, they are handled before basic multiplication and subtraction. The calculated value of \(3.24\) gives us necessary information used for further simplification of larger expressions.

Subtraction

Subtraction, represented by the 'minus' sign, involves finding the difference between numbers. In the context of our expression, it helps in determining the result after other operations have been performed. Subtraction is usually one of the final steps when simplifying expressions.
The expression we need to simplify and subtract is\(-76.14 - 3.24\).When subtracting negative numbers, it's helpful to think of it as moving to the left on a number line. Therefore, you take \(-76.14\) and move left another \(3.24\) places, resulting in\(-79.38\).This step finalizes our simplified expression, aiming to ensure that the solution is correct and fits logically into the arithmetic sequence.