Problem 9 - Elementary Algebra

Question

Use the order of operations to simplify the following. $$ 2^{3}+3^{3}-8 \cdot 4 $$

Step-by-Step Solution

Answer

Question: Simplify the expression 2^3 + 3^3 - 8 x 4 using the order of operations. Answer: 3

Step 1: Identify the Operations

In the given expression, we have three operations: Exponents (2^3 and 3^3), Multiplication (8 x 4), and Addition and Subtraction.

Step 2: Perform Exponentiation

According to order of operations (PEMDAS), we must perform exponentiation first: $$ 2^3 = 2 \cdot 2 \cdot 2 = 8 \\ 3^3 = 3 \cdot 3 \cdot 3 = 27 $$ Now, the expression becomes: $$ 8 + 27 - 8 \cdot 4 $$

Step 3: Perform Multiplication

Next in the order of operations is Multiplication. We have one multiplication operation in our expression: $$ 8 \cdot 4 = 32 $$ Substitute the result back into the expression: $$ 8 + 27 - 32 $$

Step 4: Perform Addition and Subtraction

Finally, we perform addition and subtraction from left to right: $$ 8 + 27 = 35 $$ Then, we do the subtraction: $$ 35 - 32 = 3 $$ The simplified expression is 3.

Key Concepts

Exponentiation

Exponentiation is one of the key operations in mathematics, and it comes first in the order of operations. When we exponentiate, we multiply a number by itself a specified number of times. For instance, in the expression \(2^3\), this indicates that we multiply 2 by itself three times: \(2 \cdot 2 \cdot 2\), resulting in 8. Similarly, \(3^3\) means \(3 \cdot 3 \cdot 3\), which equals 27. Understanding how to handle exponents is crucial because they can quickly change the magnitude of numbers in an expression. If you're ever unsure, break the exponentiation down into repeated multiplications, as we've done here. Make sure to address any exponents in your expression before moving on to the subsequent steps in the order.

Multiplication

After we've dealt with exponentiation, the next step in the order of operations is multiplication. Multiplication involves finding the product of two numbers. In our expression, we have the multiplication operation \(8 \cdot 4\). This can be done by adding 8 four times, or simply calculating it as \(8 + 8 + 8 + 8\), which results in 32.The beauty of multiplication is that it simplifies repeated addition into a single operation, saving time and effort. Multiplication must be completed before moving on to addition and subtraction because it has a higher precedence in the order of operations. Remembering this order can significantly impact the solution to the problem.

Addition and Subtraction

Once exponentiation and multiplication have been completed, we turn to addition and subtraction, the final steps in the order of operations. In our case, the operation remaining after evaluating exponents and multiplication is \(8 + 27 - 32\).- Start by adding \(8 + 27\), which equals 35.- Then subtract \(32\) from 35, which results in 3.Even though they are last in the order, addition and subtraction are crucial as they determine the final result. They are performed from left to right, which is important for maintaining the correct order of evaluation in complex expressions. Practice these operations to increase confidence and ensure accuracy in simplifying expressions.