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Chapter 5: Problem 27

Assume that all variables represent nonzero real numbers. $$ (-25)^{0} $$

Short Answer

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Step by step solution

01

- Recognize the Zero Exponent Rule

The zero exponent rule states that any nonzero number raised to the power of 0 is equal to 1. This can be mathematically expressed as: \[ a^{0} = 1 \] where \( a eq 0 \).
02

- Identify the Base

In the given expression, \( (-25)^{0} \), identify the base number. Here, the base is -25.
03

- Apply the Zero Exponent Rule

Apply the zero exponent rule: Since -25 is a nonzero number, according to the zero exponent rule, \( (-25)^{0} = 1 \).

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Key Concepts

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

Zero Exponent Rule
Understanding the zero exponent rule is crucial in algebra. Simply put, this rule states that any nonzero number raised to the power of zero equals 1. This is expressed as: \[ a^{0} = 1 \] where \( a eq 0 \). Whether the base is positive or negative does not matter as long as it is not zero. For example:
  • \( 5^{0} = 1 \)
  • \( -7^{0} = 1 \)
  • \( (1234)^{0} = 1 \)
The reason why this rule holds is because of how exponents work with division and multiplication. As long as the base is not zero, the rule remains consistent. If the exponent is zero, the value is always 1. This is a fundamental concept in algebra.
Exponents
Exponents are used to represent repeated multiplication of a number by itself. For example, \(2^3\) means \(2 \times 2 \times 2\). In exponents, the number being multiplied is called the base, and the number of times it is multiplied is the exponent. Key rules of exponents include:
  • Product Rule: \(a^m \times a^n = a^{m+n}\)
  • Quotient Rule: \(\frac{a^m}{a^n} = a^{m-n}\)
  • Power Rule: \((a^m)^n = a^{mn}\)
  • Zero Exponent Rule: \(a^0 = 1\)
Mastering these rules is pivotal in simplifying expressions and solving problems that involve exponents. Each rule helps in breaking down complex exponential expressions into simpler forms.
Base
The base in expressions with exponents is the number that gets multiplied by itself. It is often denoted by the letter \(a\) in a general form such as \(a^n\). In our given problem, \((-25)^0\), the number -25 is the base. When working with exponents:
  • Always identify the base before applying any exponent rule.
  • Understand that the base can be any nonzero real number.
  • The base can be positive, negative, or a fraction if the rules stay the same.
In the zero exponent rule, no matter what base (as long as it鈥檚 not zero), the expression \((base)^0 = 1\). For instance, \((-25)^0 = 1\). Remembering this point ensures accuracy when simplifying expressions involving exponents.

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Most popular questions from this chapter

Simplify each expression so that no negative exponents appear in the final result. Assume that all variables represent nonzero real numbers. $$ \frac{12 k^{-2}\left(k^{-3}\right)^{-4}}{6 k^{5}} $$

Simplify each expression so that no negative exponents appear in the final result. Assume that all variables represent nonzero real numbers. $$ \left(3 p^{-4}\right)^{2}\left(p^{3}\right)^{-1} $$

Write each number in standard notation. $$ 7.2 \times 10^{4} $$

Write each number in standard notation. $$ 5.42 \times 10^{-4} $$

Write each number in scientific notation. $$ 0.875 $$
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