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Chapter 5: Problem 28

Assume that all variables represent nonzero real numbers. $$ (-30)^{0} $$

Short Answer

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Step by step solution

01

Understand the Zero Exponent Rule

According to the zero exponent rule, any nonzero number raised to the power of 0 is equal to 1. Formally, for any nonzero real number x, we have \( x^0 = 1 \).
02

Apply the Zero Exponent Rule

Apply the zero exponent rule to the given expression. Here, the base is \(-30\), which is a nonzero real number. Therefore, \((-30)^0\) will be equal to 1.

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Key Concepts

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

Exponents
In mathematics, exponents are a way to represent repeated multiplication of a number by itself. For example, in the term \(2^3\), the number 2 is called the base, and 3 is the exponent. This expression means that 2 is multiplied by itself three times, which equals 8. Exponents are also called powers. They are a compact way to convey large numbers and operations in mathematics.
Understanding exponents is crucial because they appear in various branches of math, including algebra, geometry, and calculus. They help simplify and solve equations involving repeated multiplication.
Properties of Exponents
There are several important properties of exponents that make it easier to work with these mathematical expressions:
  • Product of Powers: When multiplying two expressions with the same base, you can add the exponents: \(a^m \times a^n = a^{m+n}\).
  • Quotient of Powers: When dividing two expressions with the same base, you can subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
  • Power of a Power: When raising an exponential expression to another power, you multiply the exponents: \((a^m)^n = a^{m \times n} \).
  • Zero Exponent: Any nonzero number raised to the power of zero is equal to 1: \(a^0 = 1\), given that \(a eq 0\).
  • Negative Exponent: A number raised to a negative exponent is the same as its reciprocal with a positive exponent: \(a^{-n} = \frac{1}{a^n}\).
These properties are rules that simplify the computation and manipulation of numbers with exponents.
Real Numbers
Real numbers include all the numbers that can be found on the number line. This includes both rational numbers (like 7 or -3/4) and irrational numbers (like \(\sqrt{2}\) or \(\pi\)).
Real numbers are fundamental in mathematics because they represent quantities and values we encounter in everyday life. Here are some key points to understand about real numbers:
  • Rational Numbers: Numbers that can be expressed as a fraction of two integers, where the denominator is not zero. Examples include 1/2, -3, and 5.
  • Irrational Numbers: Numbers that cannot be expressed as a simple fraction. They have non-repeating, non-terminating decimal expansions. Examples include \(\pi\) and \(\sqrt{2}\).
  • Integers and Whole Numbers: Whole numbers include all positive numbers without fractions, starting from 0. Integers include positive and negative whole numbers.
Real numbers are crucial for algebraic expressions, calculus, and more advanced fields of mathematics.

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Most popular questions from this chapter

Find each product. $$ (4 z-x)\left(z^{3}-4 z^{2} x+2 z x^{2}-x^{3}\right) $$

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Simplify each expression so that no negative exponents appear in the final result. Assume that all variables represent nonzero real numbers. $$ -2 m^{-1}\left(m^{3}\right)^{2} $$
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