Question

Explain the difference between \(-(-3)^{2}\) and \([-(3)]^{2}\)

Short answer

Question: Explain the difference between the given mathematical expressions: \(-(-3)^{2}\) and \([-(3)]^{2}\). Answer: The main difference between the two expressions is the placement of the negative sign. In the first expression \(-(-3)^{2}\), the exponent is calculated first, then the negative sign is applied, resulting in a negative answer (-9). In the second expression \([-(3)]^{2}\), the negative sign is applied first within the brackets, then the exponent is calculated, resulting in a positive answer (9).

Step-by-step solution

Analyzing and simplifying \(-(-3)^{2}\)

First, correctly apply the order of operations (PEMDAS/BODMAS) to the expression \(-(-3)^{2}\). This tells us to first evaluate the exponent, and then deal with the negative sign in front of the parenthesis. So, we have: $$(-3)^{2} = 9$$ Now, we need to take the negative of 9: $$-9 = -9$$ The simplified expression for \(-(-3)^{2}\) is -9.

Analyzing and simplifying \([-(3)]^{2}\)

Now, we need to apply the order of operations (PEMDAS/BODMAS) to the expression \([-(3)]^{2}\). This time, the negative sign is inside the square brackets, so we first need to evaluate the negative within the brackets: $$[-(3)] = -3$$ Now, we have \((-3)^{2}\), which we already solved in Step 1: $$(-3)^{2} = 9$$ The simplified expression for \([-(3)]^{2}\) is 9.

Comparing the simplified expressions

Now that we have simplified both expressions, we can compare their results to explain the difference between the original expressions: 1. \(-(-3)^{2}\) simplifies to -9. 2. \([-(3)]^{2}\) simplifies to 9. The main difference between the two expressions is the placement of the negative sign. In the first expression, the exponent is calculated first, then the negative sign is applied, resulting in a negative answer. In the second expression, the negative sign is applied first within the brackets, then the exponent is calculated, resulting in a positive answer.

Key concepts

Negative Exponents

Understanding negative exponents is fundamental to simplifying expressions correctly. A negative exponent indicates that the base number should be turned into its reciprocal and then raised to the absolute value of the given exponent. For example, the expression \( a^{-n} \) is equivalent to \( \frac{1}{a^n} \), where \( a \) is the base and \( n \) is the exponent.

In the context of our exercise, this concept assists in recognizing that a negative sign in front of an exponent doesn’t change the base but affects the whole expression after the exponentiation has been completed. This is a crucial distinction that influences the final sign of the answer.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form without changing their value. This often includes combining like terms, using the laws of exponents, and performing operations within parentheses. An essential aspect of simplification is to always perform operations in the correct order, which is where PEMDAS/BODMAS comes into play.

In our example, simplification is used to demonstrate how the location of a negative sign relative to parentheses and exponents can lead to different results even when the numbers involved are identical. This underscores the importance of paying close attention to the order of operations when simplifying mathematical expressions.

PEMDAS/BODMAS

PEMDAS (which stands for Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction) and BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) are acronyms that help students remember the order of operations in mathematics. They guide us through the steps needed to solve mathematical expressions accurately.

Consider the exercise \( -(-3)^{2} \) versus \( [-(3)]^{2} \). Here the correct application of PEMDAS/BODMAS is vital. The conflicting answers of -9 and 9 signify the importance of understanding and applying the correct order of operations. In the first instance, the exponent is resolved before applying the negative sign, while in the second, the negative sign inside the brackets is addressed first and then the exponent. This difference in sequence results in opposite signs for the final answer, demonstrating the value of the order of operations in simplifying expressions.