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Chapter 1: Problem 38

Evaluate each expression. See Example 6 $$ -6^{2} $$

Short Answer

Expert verified
The expression evaluates to -36.

Step by step solution

01

Understand the Problem

We need to evaluate the expression \[-6^{2}\]. It is important to recognize that the negative sign is not included in the exponentiation.
02

Exponentiate the Base

Apply the exponent to the base number, which is 6.Calculate:\[6^2 = 6 \times 6 = 36\].Notice that the negative sign is not included in this calculation.
03

Apply the Negative Sign

After calculating the square of 6 to get 36, apply the negative sign in front of the expression.Thus, \[-6^2 = -(6 \times 6) = -36\].

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Key Concepts

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

Exponentiation
Exponentiation is a mathematical operation where a number, known as the base, is multiplied by itself a specified number of times, called the exponent. For example, in the expression \(6^2\), 6 is the base and 2 is the exponent. This means you multiply 6 by itself:
  • \(6^2 = 6 \times 6 = 36\)
It's important to remember that exponentiation applies only to the base and does not automatically include any preceding negative signs. This is a common point of confusion, but it's crucial for correctly solving expressions involving exponents.
Negative Numbers
Negative numbers represent a value less than zero and are often symbolized using a minus sign (-) in front of a number. For instance, \(-6\) means six units in the opposite direction of positive values on a number line.
When dealing with expressions like \(-6^2\), it's essential to recognize that the negative sign is separate from the base and the exponent. First, compute the exponentiation by ignoring the negative sign:
  • \(6^2 = 36\)
Then, apply the negative sign after the exponentiation:
  • \(-6^2 = -36\)
This helps avoid mistakes when performing operations with negative numbers.
Order of Operations
The order of operations is a set of rules that dictates the sequence in which mathematical operations should be performed to ensure consistent results. Commonly remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), it highlights the precedence of operations.
  • First, solve expressions within parentheses.
  • Next, handle exponents such as powers and square roots.
  • Then, perform multiplication and division from left to right.
  • Finally, execute addition and subtraction from left to right.
In the exercise \(-6^2\), the order of operations requires us to first calculate the exponent \(6^2 = 36\) before applying the negative sign, resulting in \(-36\). Understanding this order helps simplify complex expressions correctly.

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Most popular questions from this chapter

In your own words, explain why every natural number is also a rational number but not every rational number is a natural number.

Use a calculator to approximate each square root. simplify the expression. Round answers to four decimal places. \(\frac{(-5.161)(3.222)}{7.955-19.676}\)

Choose the fraction (s) equivalent to the given fraction. (There may be more than one correct choice.) $$ \begin{aligned} &\frac{5}{-(x+y)} \quad \text { a. } \frac{5}{(x+y)} \quad \text { b. } \frac{-5}{(x+y)}\\\ &\frac{-1}{-(x+y)} \cdot \frac{-5}{(x+y)} \end{aligned} $$

Explain why \(-(-2)\) and \(-|-2|\) simplify to different numbers.

Evaluate each expression when \(x=9\) and \(y=-2 .\) See Example 12 $$ -3 y^{2} $$
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