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Chapter 2: Problem 37

Prep Exercise 5 Explain how to evaluate \(-8^{2}\). evaluate $$-10^{6}$$

Short Answer

Expert verified
Answer: The expressions \(-8^{2}\) and \(-10^{6}\) evaluate to -64 and -1,000,000, respectively.

Step by step solution

01

Exponentiation

Begin by raising 8 to the power of 2: $$8^{2} = 8 \times 8 = 64$$
02

Add negative sign

Now, apply the negative sign to the result of the exponentiation: $$-8^{2} = -(64) = -64$$ The expression \(-8^{2}\) evaluates to -64. Now, we will evaluate \(-10^{6}\).
03

Exponentiation

Begin by raising 10 to the power of 6: $$10^{6} = 10 \times 10 \times 10 \times 10 \times 10 \times 10 = 1,000,000$$
04

Add negative sign

Now, apply the negative sign to the result of the exponentiation: $$-10^{6} = -(1,000,000) = -1,000,000$$ The expression \(-10^{6}\) evaluates to -1,000,000.

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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 that involves raising a number, called the base, to the power of an exponent. This is written as \(a^n\), where \(a\) is the base, and \(n\) is the exponent. The exponent tells you how many times to multiply the base by itself.
  • Example: In the expression \(8^2\), the base is \(8\) and the exponent is \(2\), which means you multiply \(8\) by itself: \(8 \times 8 = 64\).
  • Higher Powers: For a higher exponent such as \(10^6\), you multiply \(10\) by itself six times: \(10 \times 10 \times 10 \times 10 \times 10 \times 10 = 1,000,000\).
Exponentiation allows for handling large numbers and computations more efficiently, transforming repeated multiplications into a more concise form. This operation is fundamental for various areas of mathematics, including algebra and calculus.
Negative Numbers
Negative numbers are numbers that are less than zero. They are often represented with a minus sign (\(-\)) in front. In math, negative numbers are integral in representing values below a certain base point, such as debts in finance.
  • Representation: The number \(-8\) stands for 8 units less than zero.
  • Operations: When working with negative numbers, remember that adding a negative number is equivalent to subtraction, and subtracting a negative number is equivalent to addition.
Using negative numbers in calculations, especially with exponentiations, can change the outcome of an operation significantly. In expressions such as \(-8^2\), the negative sign modifies the final result after the base has been raised to the power, leading to a negative value.
Mathematical Operations
Mathematical operations include addition, subtraction, multiplication, division, and exponentiation, which are the building blocks of arithmetic and algebra. These operations follow specific order and rules that help calculate and solve various math problems.
  • Order of Operations: It's important to follow the correct order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).
  • Applying Negativity: When dealing with negative numbers, the order in which you perform operations becomes even more crucial. For \(-8^2\), you perform the exponentiation \(8^2\) first, getting \(64\), and then apply the negative, resulting in \(-64\).
Understanding these operations and their proper sequencing is essential for solving equations accurately and efficiently. They allow you to manipulate expressions and find solutions to complex mathematical challenges.

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Most popular questions from this chapter

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