Problem 69 - Prealgebra

Question

In Exercises 69-80, simplify the given expression. \((0.36)(7.4)-(-2.8)^{2}\)

Step-by-Step Solution

Answer

The simplified expression is \(-5.176\).

Step 1: Evaluate the First Product

Calculate the product of \(0.36\) and \(7.4\). To do this, multiply \(0.36\) by \(7.4\): \[0.36 \times 7.4 = 2.664.\]

Step 2: Calculate the Second Term Square

Compute the square of \(-2.8\). Square \(-2.8\) to get:\[(-2.8)^2 = 7.84.\]

Step 3: Simplify the Expression

Insert the results from the previous steps into the original expression. The expression is:\[(0.36)(7.4) - (-2.8)^2 = 2.664 - 7.84.\]Simplify by subtracting to find:\[2.664 - 7.84 = -5.176.\]

Key Concepts

Multiplication of Decimals

When multiplying decimals, it's similar to multiplying whole numbers, but there is an extra step at the end. Multiply as if there were no decimals. For instance, if you're multiplying \(0.36 \times 7.4\), you would multiply 36 by 74. This calculation gives you 2664.
After performing the multiplication, count the total number of decimal places in both numbers you multiplied. For \(0.36\) (two decimal places) and \(7.4\) (one decimal place), you have a total of three decimal places.
Place the decimal point in your product (2664) so that it reflects the three decimal places, resulting in \(2.664\). This is crucial for accuracy.

• Start by ignoring the decimal places and multiply as if the numbers were integers.
• Count all decimal places from the numbers you multiplied.
• Place the decimal in the result accordingly.

Squaring Negative Numbers

Squaring a negative number involves multiplying the negative number by itself. A key rule is that a negative number squared results in a positive number. This is because multiplying two negative numbers together gives a positive result.
In the given expression, you are tasked with finding \((-2.8)^2\). To do this multiplying, compute:

• \(-2.8 \times -2.8 = 7.84\)

This isn't just a matter of calculating the numbers but also understanding that the square of any negative number will always yield a positive result. Re-examine the negative sign when encountering powers, particularly even ones.

Order of Operations

The order of operations is a fundamental concept in mathematics that dictates the correct sequence in which operations should be performed to ensure accurate results. The common acronym "PEMDAS" helps remember:

• Parentheses
• Exponents (Powers and Roots, etc.)
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In this exercise, you simplified an expression with both multiplication and exponents. Begin with calculating the exponent, \((-2.8)^2\), yielding 7.84 as described earlier. Next, handle the multiplication \(0.36 \times 7.4\) resulting in \(2.664\).
Finally, these steps allow you to simplify the final expression by subtraction: \(2.664 - 7.84 = -5.176\). The order of operations ensured every part was completed in the right sequence, leading to an accurate simplification of the expression.