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Chapter 6: Problem 56

Without a calculator, decide whether the quantities are positive or negative. $$ (-73)^{0} $$

Short Answer

Expert verified
Question: Determine whether the quantity \((-73)^{0}\) is positive, negative, or neither. Answer: The quantity \((-73)^{0}\) is positive.

Step by step solution

01

Recall the exponent of 0 property

For any non-zero number a, a^0 = 1. This means that any number raised to the power of 0 is equal to 1.
02

Apply the exponent of 0 property to the given number

The given number is \((-73)^{0}\). As per the exponent of 0 property we learned in Step 1, any non-zero number raised to the power of 0 is equal to 1. So, \((-73)^{0} = 1\).
03

Determine the sign of the result

The result from Step 2 was \((-73)^{0} = 1\). The number 1 is a positive number. So, the quantity is positive.

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Key Concepts

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

Exponent of 0 property
In mathematics, understanding the Exponent of 0 property is crucial for simplifying expressions and solving problems accurately. This concept tells us that any non-zero number raised to the exponent 0 equals 1.
This rule can be represented as follows:
  • If \(a\) is any non-zero number, then \(a^0 = 1\).
A common misconception is that the sign or size of the base number affects this rule, but it doesn't. Whether you have a large positive number like \(1000^0\) or a negative one like \((-73)^0\), the result remains 1.
Utilizing the Exponent of 0 property can simplify complex expressions and aid in solving algebraic equations, making it a foundational tool in mathematics.
Positive and negative numbers
Understanding positive and negative numbers is vital in algebra and everyday math situations. Positive numbers are those greater than zero, such as 1, 2, or 100, while negative numbers are less than zero, such as -1, -2, or -100.
In our exercise, despite the base being a negative number \((-73)\), applying the exponent of 0 gives us a result of 1, which is a positive number.
This illustrates that the nature of positive and negative numbers can sometimes be altered by operations such as exponentiation. Generally, positive results are those above zero, providing a sense of increase or gain, whereas negative results indicate a decrease or loss.
Algebraic expressions
Algebraic expressions are combinations of numbers, variables, and operations like addition, subtraction, multiplication, division, and exponentiation. These help in forming equations and solving mathematical problems.
When working with algebraic expressions, understanding exponentiation and its properties, such as the exponent of 0, is key. You might see expressions like \(x^0\), and knowing it equals 1 can simplify your work.
  • Algebraic expressions can involve complex interactions of multiple terms.
  • Using rules like the exponent of 0 property allows simplification.
Expressions simplify into more manageable pieces, aiding in solving equations or understanding relationships between variables. Mastering these concepts strengthens your algebra skills and boosts problem-solving efficiency.

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Most popular questions from this chapter

Write each expression as a power raised to a power. There may be more than one correct answer. $$ 5^{2 y} $$

$$ \text { If } 3^{x}=y, \text { express } 3^{x+2} \text { in terms of } y $$

Write each expression without parentheses. Assume all variables are positive. $$ \left(\frac{2 p}{q^{3}}\right)^{5} $$

Write the expression in the form \(x^{n}\), assuming \(x \neq 0\) $$ \frac{x \cdot x^{6}}{\left(x^{3}\right)^{2}} $$

Combine radicals, if possible. \(2 \sqrt{3}+\frac{\sqrt{3}}{2}\)
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