Problem 51 - Fundamentals of Mathematics

Question

Find each value. Check each result with a calculator. \((21-3) \cdot(6-1) \cdot(7)+4(6+3)\)

Step-by-Step Solution

Answer

The final result is 666.

Step 1: Simplify Inside Parentheses

Begin by simplifying the expressions inside each set of parentheses. For \( (21-3) \), subtract to get 18; for \( (6-1) \), subtract to get 5; and for \( (6+3) \), add to get 9. The expression now becomes \[ 18 imes 5 imes 7 + 4 imes 9 \]

Step 2: Perform Multiplication

Next, perform the multiplication for each group. Start with the first part:\[ 18 imes 5 = 90 \]Then multiply the result by 7:\[ 90 imes 7 = 630 \]Now do the multiplication for the second part:\[ 4 imes 9 = 36 \]

Step 3: Add the Results

Finally, add the results from the previous step together: \[ 630 + 36 = 666 \]

Step 4: Verify with Calculator

Double check the computation using a calculator. Input the original expression: \[ (21-3) imes (6-1) imes 7 + 4 imes (6+3) \]Ensure that the final result matches: 666.

Key Concepts

Multiplication

Multiplication is one of the four basic operations in mathematics. It involves combining equal groups. When you multiply, you are basically finding the total number of items when items are grouped together in equal sizes. In the given exercise, you'll encounter multiplication like this:

• First, you multiply 18 by 5, which equals 90.
• Then, the result, 90, is multiplied by 7 to make 630.
• Additionally, 4 is multiplied by 9 to get 36.

You can think of multiplication as repeated addition. For example, multiplying 7 by 5 is the same as adding 7 five times: 7 + 7 + 7 + 7 + 7.
Multiplication can be used to simplify larger expressions. Rather than add the same number multiple times, multiplication allows you to use these numbers in a quicker format. After we've made all necessary calculations involving multiplication, we have all components ready for the next fundamental operation: addition.

Addition

Addition is used to calculate the total or sum of numbers. It's another basic operation that plays a crucial role in arithmetic. Following the multiplication steps in the exercise, the results are added to achieve the final value.
In the exercise example, we perform all the multiplications first to get:

• 630, resulting from 18 multiplied by 5 and then by 7.
• 36, resulting from 4 multiplied by 9.

These individual results are then added together using addition:

• \(630 + 36\)

This final addition gives us the complete and correct answer: 666. Addition is the final step that assembles all calculated values together, emphasizing the importance of following the correct order of operations.

Parentheses Simplification

Parentheses are used in mathematical expressions to group parts of the expression that should be simplified first. These groupings are very important as they dictate the order of execution in solving a mathematical problem. In the provided exercise, there are several operations inside parentheses that need to be resolved before performing multiplication and addition.
When simplifying within parentheses, you resolve each small calculation inside independently:

• Subtract 3 from 21, obtaining 18.
• Subtract 1 from 6, resulting in 5.
• Add 6 and 3 together to get 9.

After these calculations, the expression simplifies and looks much cleaner, setting us up for the multiplication steps.
By following the parentheses simplification rules properly, we ensure that the order of operations is respected, leading to the accurate calculation of the entire expression.