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Chapter 4: Problem 9

Decide whether each expression is equal to \(0,1,\) or \(-1 .\) $$ (-4)^{0}-4^{0} $$

Short Answer

Expert verified
The expression is equal to 2.

Step by step solution

01

- Evaluate \((-4)^{0}\)

Any non-zero number raised to the power of 0 equals 1. Therefore, \((-4)^{0} = 1\).
02

- Evaluate \(-4^{0}\)

When evaluating \(-4^{0}\), apply the exponentiation first. Since \(4^{0} = 1\), \(-4^{0} = -1\).
03

- Subtract the results from Step 1 and Step 2

Subtract the results obtained: \(1 - (-1) = 2\).

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Key Concepts

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

Exponentiation Rules
Exponentiation mainly involves raising a number, known as the base, to a certain power, known as the exponent. The rules are straightforward but essential to master:
  • When any number is raised to the power of 0, the result is always 1, as long as the base is not zero. This is shown in the exercise where \((-4)^{0} = 1\).

  • When raising a negative number to an exponent, attention is needed to parentheses. If there's no parenthesis, the exponent only applies to the number, not the negative sign.

For example, in the exercise, \(-4^{0}\) is treated differently because the exponent applies only to 4.
Negative Exponents
Understanding negative exponents is crucial. A negative exponent means that the base is reciprocated (flipped to its inverse). Here are the main points:
  • For any non-zero value \(a\), \(a^{-n} = \frac{1}{a^{n}}\).

  • This law indicates that raising a number to a negative exponent results in a fraction.

Consider the expression \(-4^{0}\) from the exercise; however, it simplifies to -1, not using negative exponents, but it's essential for broader understanding.
Base Powers
Understanding the base when applying powers determines the right interpretation of the expression:
  • A base with an exponent applies the exponent to the entire base, while a negative sign outside parentheses is handled differently.

  • In the helpful exercise \((-4)^{0} = 1\) and \(-4^{0} = -1\), shows this difference. The parentheses mean the negative sign is part of the base, while without them, it is not.

This simple rule avoids common mistakes when solving exponential expressions.

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Most popular questions from this chapter

Astronomers using the Spitzer Space Telescope discovered a twisted double- helix nebula, a conglomeration of dust and gas stretching across the center of the Milky Way galaxy. This nebula is \(25,000\) light-years from Earth. If one light-year is about \(6,000,000,000,000\) (that is, 6 trillion) miles, about how many miles is the twisted double-helix nebula from Earth? (Source: hutp:l/ articles.news.aol.com)

Subtract. See Example \(\boldsymbol{\delta}\) $$ \begin{array}{r} {-6 t^{3}+4 t^{2}} \\ {8 t^{3}-6 t^{2}} \end{array} $$

Evaluate. $$ 100(72.79) $$

Evaluate each expression for \(x=3 .\) See Sections 1.3 and 1.6. $$ 3 x^{2}-3 x+4 $$

Use scientific notation to calculate the answer to each problem. Write answers in scientific notation. $$ \frac{0.000000081(0.000036)}{0.00000048} $$
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