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Chapter 4: Problem 29

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

Short Answer

Expert verified
The value is 1.

Step by step solution

01

Understand the Zero Exponent Rule

The zero exponent rule states that any nonzero number raised to the power of zero equals 1. Mathematically, this is written as a^0 = 1 , where a is a nonzero real number.
02

Apply the Zero Exponent Rule

In this exercise, the base is -25, which is a nonzero real 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
The zero exponent rule is an important concept in algebra. It states that any nonzero number raised to the power of zero is equal to 1. This can be mathematically written as,\[a^0 = 1\],where \( a \) is any nonzero real number.
For example, consider \((-25)^0\). Although the base of -25 might look intimidating, applying the zero exponent rule simplifies it:
\[(-25)^0 = 1\].
exponents in algebra
Exponents in algebra help to express repeated multiplication. For example, \(a^3\) means \(a\) multiplied by itself three times: \(a \times a \times a\).
There are a few important rules for working with exponents:
  • 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: \(a^0 = 1\), where \(a\) is a nonzero real number
Among these, the zero exponent rule is particularly useful because it simplifies any expression where a nonzero base has an exponent of zero to 1.
evaluating exponents
Evaluating exponents involves using rules such as the zero exponent rule to simplify expressions. Let鈥檚 consider some steps for evaluating exponents:
  • Identify the base and the exponent.
  • Apply the relevant exponent rule.
  • Simplify the expression.
For instance, in the given exercise:
- Identify the base as -25 and the exponent as 0.
- Apply the zero exponent rule: \((-25)^0 = 1\).
This simple approach makes it easy to evaluate expressions with exponents.

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Most popular questions from this chapter

Perform the indicated operations. $$ \left[-\left(4 m^{2}-8 m+4 m^{3}\right)-\left(3 m^{2}+2 m+5 m^{3}\right)\right]+m^{2} $$

Perform the indicated operations. Subtract \(-\left(-4 x+2 z^{2}+3 m\right)\) from \(\left[\left(2 z^{2}-3 x+m\right)+\left(z^{2}-2 m\right)\right]\)

Perform the indicated operations. $$ [2 p-(3 p-6)]-[(5 p-(8-9 p))+4 p] $$

Let \(f(x)=x^{2}-9, g(x)=2 x,\) and \(h(x)=x-3 .\) Find each of the following. $$ (f g)(0) $$

Write using only positive exponents and then evaluate. Assume that all variables represent nonzero real numbers. $$ \left(\frac{5}{4}\right)^{-2} $$
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