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Chapter 1: Problem 80

Simplify each expression. $$[2(-3)]^{2}$$

Short Answer

Expert verified
The simplified expression is 36.

Step by step solution

01

Solve the bracket operation

First, calculate the product in the brackets which is \(2(-3)\) resulting in -6.
02

Apply the exponent operation

Next, square the result obtained in step 1. This means multiplying -6 by itself. So, \[(-6)^{2} = 36\].
03

Write the final answer

Having solved the expression, the result is 36.

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Key Concepts

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

Exponents
Exponents are a way of expressing repeated multiplication of the same number. For example, when we write \(a^2\), it means \(a\) multiplied by itself: \(a \times a\). In our exercise, the exponent is 2, indicating that we must square the base number. After evaluating the expression in the brackets, we need to square the resulting number, \(-6\).
To perform this step, multiply \(-6\) by itself, which results in \(-6 \times -6 = 36\). Notice that squaring a negative number results in a positive number because a negative number multiplied by itself produces a positive product.
Some important points to remember about exponents are:
  • Any number raised to the power of 0 is 1, except for 0 itself.
  • Raising a negative number to an even power results in a positive number.
  • Raising a negative number to an odd power results in a negative number.
Order of Operations
The order of operations is crucial when simplifying expressions to ensure that everyone solves it in the same manner and arrives at the same answer. The standard order of operations is remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).
In our exercise, we first perform the operation inside the brackets \([2(-3)]\), then apply the exponent to the result. This follows the order of operations:
  • Parentheses: We calculate \(2 \times -3\) first.
  • Exponents: Next, we square the result \(-6\).
Following PEMDAS ensures you handle more complex expressions correctly by breaking them into manageable steps.
Negative Numbers
Working with negative numbers can sometimes be tricky, but with practice, it becomes intuitive. A negative number is any number less than zero, and these numbers behave differently under multiplication and division.
When multiplying two negative numbers, the result is positive. This is why in our exercise, squaring \(-6\) resulted in \(36\), as two negatives multiplied together produce a positive.
  • Multiplying Negatives: \((-a) \times (-b) = ab\)
  • Multiplying a Positive and a Negative: Results in a negative number, \(a \times (-b) = -ab\)
Understanding these rules helps you correctly simplify expressions and solve equations involving negative numbers. Remember, always apply these basic rules, and practice often to master the nuances of negative numbers.

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