Problem 50 - Algebra A Combined Function

Question

Simplify each expression. 2^{3}-6 \cdot 3$

Step-by-Step Solution

Answer

The simplified expression is -10.

Step 1: Simplify the exponent

First, we need to simplify the expression with the exponent. Calculate \(2^3\): \[2^3 = 2 \times 2 \times 2 = 8\] So the expression becomes \[8 - 6 \cdot 3\]

Step 2: Perform the multiplication

Next, we simplify the multiplication part of the expression. Multiply: \(6 \cdot 3\): \[6 \times 3 = 18\] Now the expression is: \[8 - 18\]

Step 3: Perform the subtraction

Finally, subtract the result from the multiplication from the result of the exponent: Calculate \(8 - 18\): \[8 - 18 = -10\] Thus, the simplified expression is \(-10\).

Key Concepts

Exponents

Exponents are a way to express repeated multiplication of a number by itself. In the exercise, the expression includes an exponent: \(2^3\). This means you need to multiply 2 by itself a total of three times. It's like stacking the same number on top of itself multiple times.
For example:

• \(2^3 = 2 \times 2 \times 2\)

The product of these multiplications is 8, making the expression: \(8 - 6 \cdot 3\). Understanding how to handle exponents is crucial, as they simplify expressions and make calculations faster. Always ensure you complete exponents before moving to other operations to follow the correct order of operations.

Order of Operations

In mathematics, correctly simplifying expressions relies on completely understanding the 'order of operations'. It's like having a set of rules to ensure calculations are done in the right order.
The common acronym PEMDAS helps remember the sequence:

• P: Parentheses
• E: Exponents
• M/D: Multiplication and Division (from left to right)
• A/S: Addition and Subtraction (from left to right)

In the given expression, \(2^3 - 6 \cdot 3\), you start with calculating the exponent before moving to multiplication and finally subtraction. Sticking to PEMDAS ensures that you arrive at the correct outcome, maintaining accuracy in operations.

Multiplication

Multiplication is one of the basic arithmetic operations, integral in expression simplification. It involves taking a number and adding it to itself a certain number of times.
In the step-by-step solution, we multiply 6 and 3, represented by \(6 \cdot 3\). This multiplication can be thought of as:

• \(6 + 6 + 6 = 18\)

After calculating the exponent \(2^3\), the subsequent operation is multiplication. By calculating \(6 \cdot 3 = 18\), the expression then becomes \(8 - 18\). Understanding multiplication is vital to simplifying expressions, setting you up nicely for the final operations.

Subtraction

Subtraction is the process of taking away a number from another, effectively reducing the total value. After simplifying the exponents and performing the multiplication, the final step involves subtraction.
In our expression, once we have \(8 - 18\), we need to subtract 18 from 8. Imagine it as a balance where you're reducing the value:

• Starting with 8 (from the exponent result)
• Take away 18 (from the multiplication result)
• The outcome is \(-10\)

Realizing that subtraction can yield negative results is crucial, so here we find that the initial number is less than the one being subtracted, resulting in \(-10\). Effective subtraction completes the process of simplification.