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Chapter 1: Problem 47

Simplify. $$ [2 \cdot(5-8)]^{2}-12 $$

Short Answer

Expert verified
24

Step by step solution

01

Evaluate the expression inside the parentheses

Start by solving the expression inside the parentheses: \(5-8\).
02

Perform the subtraction

Calculate \(5-8 = -3\).
03

Multiply by 2

Now, multiply the result by 2: \(2 \times (-3) = -6\).
04

Apply the exponentiation

Raise the result to the power of 2: \((-6)^{2} = 36\).
05

Subtract 12

Finally, subtract 12 from 36: \(36 - 12 = 24\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Order of operations
In algebra, it's essential to follow the order of operations to correctly simplify expressions. This order is often remembered by the acronym PEMDAS, which stands for:
  • Parentheses
  • Exponents
  • Multiplication
  • Division
  • Addition
  • Subtraction
Always start with parentheses, then handle exponents. Next, perform any multiplication or division from left to right. Finally, handle addition or subtraction from left to right. By following this order, you ensure calculations are accurate and consistent.
Parentheses
Parentheses are used to group parts of an expression that should be evaluated first. In our exercise, we start with the expression inside the parentheses: \(5-8\). This is crucial because it can change the result drastically. Without proper evaluation, you could end up with the wrong number. Simplify inside the parentheses before moving on to other operations.
Exponentiation
Exponentiation is the process of raising a number to a power. In our example, after simplifying inside parentheses and doing the multiplication, we get \(-6\). We then apply the exponentiation: \((-6)^{2} \). Remember, raising a negative number to an even power results in a positive number, while raising it to an odd power results in a negative number. Thus, \((-6)^{2} = 36\).
Subtraction
Subtraction is the process of taking one number away from another. In the final step of our exercise, we subtract 12 from 36. It's straightforward but crucial for getting the correct simplified result.

Always be careful with negative signs throughout the process, as they can change the final outcome.

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