Problem 26 - Fundamentals of Mathematics

Question

Use the order of operations to determine each value. $$2^{3}-2 \cdot 4 $$

Step-by-Step Solution

Answer

The value is 0.

Step 1: Evaluate the Exponent

The expression is given as \(2^3 - 2 \cdot 4\). According to the order of operations, we first solve the exponentiation. \(2^3\) means \(2\) multiplied by itself three times: \(2 \times 2 \times 2 = 8\).

Step 2: Perform Multiplication

Next, handle the multiplication part of the expression \(- 2 \cdot 4\). Multiply \(2\) by \(4\) to get \(8\). So, we transform the expression to \(8 - 8\).

Step 3: Perform Subtraction

Now, perform the subtraction operation on the expression \(8 - 8\). Subtract \(8\) from \(8\) to get \(0\).

Key Concepts

Exponentiation

Exponentiation is an essential concept in mathematics that allows us to express how many times a number, called the base, is multiplied by itself. In this context, the base is 2, and the exponent is 3, represented as \(2^3\). Exponentiation is generally one of the first operations to perform when dealing with any algebraic expression, due to the order of operations (PEMDAS/BODMAS - Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction).

For \(2^3\), we calculate it as:

• 2 \(\times\) 2 = 4
• 4 \(\times\) 2 = 8

So, \(2^3\) equals 8. By understanding exponentiation, you can easily simplify complex expressions and solve problems that involve powers of numbers.

Multiplication

Multiplication is a fundamental arithmetic operation that represents repeated addition. In the expression we are solving, we have \(- 2 \cdot 4\), which requires applying the multiplication operation.

To solve the multiplication:

• Multiply 2 by 4.
• This calculation yields 8.

The order of operations dictates that multiplication is performed before addition and subtraction, so it's important to complete this step before proceeding to the next operation in your expression.

Once solved, we substitute back into the expression, having reduced it to \(8 - 8\). Mastering multiplication can simplify many mathematical problems and ensure accuracy in solving expressions.

Subtraction

Subtraction is the process of deducting one number from another. It is the final operation performed in the order of operations for our expression \(8 - 8\). Subtraction involves taking away a quantity to find the difference between numbers.

Let's perform the subtraction:

• Start with the number 8.
• Subtract 8 from it, which means you take away 8.
• The result is 0.

Although it is the last operation you perform when following the order of operations, subtraction is a critical skill in problem-solving across various mathematical contexts, from basic arithmetic to more complex algebraic calculations.