Problem 84 - Prealgebra

Question

Given \(\mathrm{a}=3.3, \mathrm{~b}=7.3\), and \(\mathrm{c}=3.4\), evaluate the expression \(\mathrm{ab}-\mathrm{c}^{2}\).

Step-by-Step Solution

Answer

The evaluated expression is 12.53.

Step 1: Calculate the Value of ab

First, we need to calculate the product of \(a\) and \(b\). Given that \(a = 3.3\) and \(b = 7.3\), the calculation is as follows: \[ ab = 3.3 \times 7.3 = 24.09. \]

Step 2: Calculate the Value of c^2

Next, we find the square of \(c\). Given \(c = 3.4\), the calculation is \[ c^2 = 3.4^2 = 11.56. \]

Step 3: Evaluate ab - c^2

Now, substitute the calculated values into the expression \(ab - c^2\). We know \(ab = 24.09\) and \(c^2 = 11.56\). Thus, we perform the subtraction: \[ ab - c^2 = 24.09 - 11.56 = 12.53. \]

Key Concepts

Understanding Multiplication

Multiplication is one of the basic arithmetic operations, essential for combining numbers into a larger single value through repeated addition. When you multiply two numbers, you're essentially adding one number to itself as many times as the other number specifies.

For example, in the problem, we have two numbers: \( a = 3.3 \) and \( b = 7.3 \). To calculate \( ab \) or \( a \times b \), you multiply them together. This is similar to saying, "take 3.3 and add it to itself 7.3 times." Here, the result of \( 3.3 \times 7.3 \) is \( 24.09 \).

Here are a few tips for multiplication:

• Establish a solid understanding of basic multiplication facts; this helps speed up calculations.
• Break down complex numbers into simpler parts if multiplication gets overwhelming.
• Use the distributive property for complex multiplication, such as breaking numbers into whole and decimal parts. This can simplify the process when dealing with decimals.

Exploring Exponents

Exponents are a way to represent repeated multiplication of a number by itself. The expression \( c^2 \) means "multiply \( c \) by itself." This is different from simple multiplication because the same number is used as both factors repeatedly.

In the given exercise, you're asked to find \( c^2 \) where \( c = 3.4 \). Calculating \( 3.4^2 \) involves multiplying 3.4 by itself, which equals \( 11.56 \).

Here are some key points about exponents:

• The base number is the number being multiplied (for example, \( c \) in \( c^2 \)).
• The exponent indicates how many times to use the base as a factor. In \( c^2 \), the exponent is 2, so we multiply \( c \) by itself.
• Exponents can change the value of a number significantly. Always take care to calculate them accurately, especially when dealing with higher powers.

Mastering Subtraction

Subtraction is another fundamental arithmetic operation, used to find the difference between numbers. Essentially, subtraction is the process of removing a certain amount from a number, leaving you with the remainder.

In the context of the problem, you first found the values for \( ab \) and \( c^2 \) as \( 24.09 \) and \( 11.56 \) respectively. The problem then asks for the difference \( ab - c^2 \), which is \( 24.09 - 11.56 = 12.53 \).

To become proficient in subtraction:

• Line up numbers by their decimal points when dealing with decimals, which can help prevent mistakes.
• Start subtracting from the rightmost side (units place) towards the left, managing any necessary borrowing methodically.
• Check your work by adding the difference to the smaller number to ensure it equates to the larger number. This is a helpful verification step after performing subtraction.