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Chapter 3: Problem 30

Simplify. $$-(-7)^{2}$$

Short Answer

Expert verified
-49

Step by step solution

01

Understand the problem

The expression \(-(-7)^{2}\) needs to be simplified. Identify the order of operations required.
02

Evaluate the exponent

First, calculate the square of \(-7\). Using the formula for squares, we have \((-7)^{2} = (-7) \times (-7) = 49\).
03

Apply the negative sign

Now, apply the negative sign to the result from Step 2. Thus, \(-49 = -(49)\).

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Key Concepts

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

Order of Operations
The **Order of Operations** is crucial to correctly solving mathematical expressions.
It is often remembered using the acronym **PEMDAS** (Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)).
When simplifying an expression, follow this sequence:
  • **Parentheses**: Always start with operations inside parentheses.
  • **Exponents**: Next, handle any exponents or powers.
  • **Multiplication and Division**: Perform these operations from left to right.
  • **Addition and Subtraction**: Perform these last, also from left to right.
In the given problem \(-(-7)^{2}\), we follow the order:
  • First, handle the exponent \((-7)^{2}\).
  • Then, apply the negative sign in front.
This ensures correct and consistent results.
Exponents
**Exponents** indicate how many times a number (the base) is multiplied by itself.
In the expression \(-(-7)^{2}\), \(-7\) is raised to the power of 2.
This is expressed as \((-7) \times (-7)\):
  • First, multiply the negative number \(-7\) by itself.
  • The product of two negative numbers is positive.
Therefore, \((-7) \times (-7) = 49 \). Exponents are handled before other operations except for parentheses in the order of operations. This often can change the result dramatically.
Negative Numbers
**Negative Numbers** are numbers less than zero and they have unique rules when it comes to arithmetic operations.
  • Multiplying or dividing two negative numbers results in a positive number.
  • Multiplying or dividing a positive number by a negative number results in a negative number.
  • Adding or subtracting negative numbers follows the rules for addition and subtraction with consideration for direction on the number line.
In our example, after evaluating \((-7)^{2} = 49\),
we apply the outer negative sign:
Hence, \-49\. Knowing how negative signs interact with other operations is key to simplifying expressions accurately.

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