Problem 28 - Algebra A Combined Function

Question

Simplify each expression. $$ 6-2 \cdot 2+2^{5} $$

Step-by-Step Solution

Answer

The simplified expression is 34.

Step 1: Simplify the Exponentiation

First, simplify the exponentiation in the expression. We have \(2^5\) which means \(2\) multiplied by itself 5 times.\[2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32\]So, the expression now becomes:\[6 - 2 \cdot 2 + 32\]

Step 2: Perform Multiplication

Next, perform the multiplication operation in the expression. We have \(2 \cdot 2\).\[2 \cdot 2 = 4\]Now, substitute this back into the expression:\[6 - 4 + 32\]

Step 3: Perform Addition and Subtraction from Left to Right

Finally, carry out the addition and subtraction from left to right.1. Subtract first: \[ 6 - 4 = 2 \] The expression is now: \[2 + 32\]2. Add next: \[ 2 + 32 = 34 \]Thus, the simplified expression is \(34\).

Key Concepts

Exponentiation

Understanding exponentiation is key to solving expressions that include powers. When you see a base number raised to an exponent, like in the term \(2^5\), it means you multiply the base (2 in this case) by itself, as many times as the exponent indicates.
This can be thought of as repeated multiplication. For \(2^5\), it translates to \(2 \times 2 \times 2 \times 2 \times 2\), which equals 32.

• **Base Number**: The number being multiplied (2).
• **Exponent**: Number of times to multiply the base (5).

After calculating the exponentiation, substitute this result back into the original expression before solving any other operations.

Multiplication

Once exponentiation is simplified, the next step involves addressing any multiplication within the expression. Using the order of operations, multiplication needs to be performed before any addition or subtraction.
In our example, we deal with the expression \(2 \cdot 2\), which equals 4. Multiplication is essentially repeated addition. Here, we are adding the number 2 twice.

• **Operation Sequence**: Always do multiplication before moving onto addition or subtraction in an expression.

Replacing the multiplication result back into the expression, it updates to \(6 - 4 + 32\). Solving multiplication early helps simplify expressions step-by-step.

Addition and Subtraction

Finally, with exponentiation and multiplication resolved, we focus on addition and subtraction. According to order of operations, these should be handled from left to right, just like reading a sentence.
Starting with the simplified expression \(6 - 4 + 32\), we perform the subtraction first: \(6 - 4 = 2\). This gives us \(2 + 32\).

• **Left-to-Right Rule**: Work through the expression starting from the leftmost operation.
• **Final Addition**: Lastly, add \(2 + 32\) to get the final answer of 34.

Thinking left-to-right ensures you follow the correct order using both addition and subtraction correctly, leading to the final simplified result.