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Chapter 1: Problem 19

Simplify the given expressions. Express results with positive exponents only. $$(8 a)^{0}$$

Short Answer

Expert verified
The simplified expression is 1.

Step by step solution

01

Understand the zero exponent rule

The zero exponent rule states that any non-zero number or algebraic expression raised to the power of zero equals one. This applies universally, meaning \((a)^0 = 1\) as long as \(a\) is not zero.
02

Apply the zero exponent rule

In the given expression \((8a)^{0}\), both 8 and \(a\) are within the parentheses and raised to the power of zero. By applying the zero exponent rule, we can directly say that \((8a)^{0} = 1\).

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Key Concepts

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

Algebra
Algebra is a cornerstone of mathematics that deals with symbols and the rules for manipulating these symbols. In simpler terms, it's all about figuring out ways to solve equations and understand relationships between quantities. You'll often see letters and numbers mixed together in algebraic expressions, and your goal is to make sense of them.

Here's a quick look at some key ideas:
  • Expressions: These are combinations of numbers, variables, and operations. For example, \(8a\) is an expression involving a number (8) and a variable (a).
  • Equations: These represent a statement that two expressions are equal. When you solve an equation, you're finding the value of the variable that makes the equation true.
  • Rules and Properties: Algebra relies heavily on rules and properties, like the zero exponent rule, to simplify and solve problems.
Algebra isn't just about crunching numbers; it's a way to think logically about everyday problems and find solutions through clear steps and expressions.
Exponents
Exponents are a shorthand way of expressing how many times a number (the base) is multiplied by itself. They're a fundamental part of algebra and drive many calculations in mathematics. Understanding exponents is crucial because they show up everywhere, from calculating compound interest to measuring the exponential growth of a population.

There are several key points to grasp about exponents:
  • Positive Exponents: Indicate how many times to multiply the base by itself. For example, \(3^2 = 3 \times 3 = 9\).
  • Zero Exponents: As seen in the problem, anything (except zero) raised to the zeroth power is 1. This includes complex combinations like \( (8a)^0 = 1\).
  • Negative Exponents: Represent reciprocal operations. While not covered in this exercise, it's important to know that \( a^{-n} = \frac{1}{a^n} \).
Using these properties effectively can help you simplify expressions and solve algebraic problems with ease.
Mathematics Education
Mathematics education is about fostering an understanding of concepts like algebra and exponents among students. The ultimate aim is to make math accessible and enjoyable, emphasizing why mathematical rules, like the zero exponent rule, matter in real-world applications.

Effective strategies in teaching include:
  • Relating Math to Real Life: Showing how algebraic ideas apply to everyday problems helps students see the value of what they learn.
  • Breaking Down Complex Ideas: Simplifying ideas into digestible parts helps students slowly build up their understanding.
  • Interactive Learning: Engaging students with hands-on activities or technological tools can make abstract concepts more tangible.
By emphasizing these educational strategies, students can better understand and appreciate the relevance of mathematics in their daily lives, leading to stronger analytical thinking skills.

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Most popular questions from this chapter

Solve the given problems. Water flows from a fire hose at a rate of \(85 \mathrm{gal} / \mathrm{min}\). Convert this to liters per second.

Solve the given problems. Refer to Appendix B for units of measurement and their symbols. Describe the values of \(x\) and \(y\) for which (a) \(|x+y|=|x|+|y|\) and (b) \(|x-y|=|x|+|y|\).

Perform the indicated multiplications. Explain how you would perform \((x+3)^{5}\). Do not actually do the operations.

Find the indicated values. A formula used in determining the total transmitted power \(P_{t}\) in an AM radio signal is \(P_{t}=P_{c}\left(1+0.500 m^{2}\right) .\) Find \(P_{c}\) if \(P_{t}=680 \mathrm{W}\) and \(m=0.925\)

Perform the indicated operations and attach the correct units to your answers. $$\left(7.25 \frac{\mathrm{m}}{\mathrm{s}^{2}}\right)\left(\frac{1 \mathrm{ft}}{0.3048 \mathrm{m}}\right)\left(\frac{60 \mathrm{s}}{1 \mathrm{min}}\right)^{2}$$
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