Problem 44 - Prealgebra

Question

Perform the following operations according to the rule for order of operations. $$3.6+15(2.1)^{2}$$

Step-by-Step Solution

Answer

The result of the operation is 69.75.

Step 1: Evaluate the Exponent

First, calculate the expression inside the parentheses and then raise the number to the power indicated. So, calculate \((2.1)^2\). This equals \(2.1 \times 2.1 = 4.41\).

Step 2: Multiply

Now multiply the result of the exponentiation by 15. So, calculate \(15 \times 4.41\). This equals \(66.15\).

Step 3: Addition

Finally, add the original 3.6 to the result from the multiplication step. So, \(3.6 + 66.15 = 69.75\).

Key Concepts

Exponentiation

Exponentiation is the process of raising a number to the power of another number. It might seem intimidating, but it's straightforward when you break it down. When you see something like \[(2.1)^2\] it means you are taking the number 2.1 and multiplying it by itself. Essentially, it is \[2.1 \times 2.1\].
Here are some key points to remember:

• The base is the number being multiplied by itself, like 2.1 in our example.
• The exponent tells you how many times the base is used as a factor in the multiplication.
• In \[(2.1)^2\], the superscript 2 is the exponent, which means you multiply 2.1 by 2.1.
• Exponentiation is performed before multiplication and addition when following the order of operations rules.

Remember: For the base 2.1 raised to the power of 2, the result is 4.41. It's like multiplying 2.1 by itself to solidify the foundation of our other arithmetic operations.

Multiplication

Multiplication follows after exponentiation in the order of operations. When we multiply, we essentially add a number to itself a certain number of times. For example, in the equation\[15 \times 4.41\], here’s what’s happening:

• You take the result from your exponentiation, which is 4.41.
• Multiply this number by 15.
• This multiplication means you take 4.41 fifteen times and sum it up.

So,\[15 \times 4.41 = 66.15\].
Remember this technique for multiplying larger numbers:

• Break down larger numbers into smaller, more manageable numbers if needed.
• Perform multiplication in steps if dealing with decimals feels challenging.
• Always multiply before you add according to the order of operations.

Breaking down multiplication into these digestible steps helps make it easier to tackle, especially when it involves decimals like in our problem.

Addition

Addition is usually the final step when performing operations with different arithmetic methods. When we reach addition, we’ve usually cleared the other operations first. Here's how it works in our example:

• Start with the result from all earlier operations, which here is the multiplication result of 66.15.
• Add 3.6 to this number.
• This involves summing 66.15 and 3.6 to get a total.

Performing this comes down to aligning the decimal points and adding as you would whole numbers:

1. Think of it in terms of lining up place values (units under units, tenths under tenths).
2. Start from the rightmost column and move to the left, adding each column and carrying over any amounts when the sum exceeds 9.

So,\[ 66.15 + 3.6 = 69.75 \].Never forget that in the sequence based on the order of operations, addition comes after you have completed all exponentiation and multiplication tasks, neatly bringing all your calculations to a conclusion.