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Chapter 5: Problem 51

Simplify. $$ 5 x^{0} \text { when } x=-4 $$

Short Answer

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Step by step solution

01

- Understand the Zero Exponent Rule

The zero exponent rule states that any non-zero number raised to the power of zero is equal to 1. Mathematically, this is expressed as \[ x^0 = 1 \] as long as \( x eq 0 \).
02

- Apply the Zero Exponent Rule

Using the zero exponent rule, we know that \(x^0 = 1\). Since the problem is \( 5 x^0 \), first simplify \( x^0 \): \[ x^0 = 1 \]
03

- Simplify the Expression

Now, substitute \( x^0 = 1 \) into the expression \( 5 x^0 \): \[ 5 x^0 = 5 \times 1 = 5 \]

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Key Concepts

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

Exponent Rules
To understand the problem, you first need to know the basic exponent rules. One of the key rules is the Zero Exponent Rule. This rule states that any non-zero number raised to the power of zero equals 1. It's important because it simplifies expressions containing zero exponents.

For example, according to this rule: \[ a^0 = 1 \] This holds true for any non-zero value of \(a\).

Remember, the key is that the base cannot be zero. If you stumble upon \(0^0\), it's indeterminate and a different story altogether. Focus on cases where the base is a non-zero number.
Simplification
Simplification is about making an expression easier to manage or understand. Using the Zero Exponent Rule can help simplify your algebraic expression like in the given example.

Starting with the original problem: \[ 5 x^0 \text{ when } x = -4 \] Apply the Zero Exponent Rule, where \( x^0 = 1 \), regardless of \( x\). So the expression now looks like: \[ 5 \times 1 \] which simply equals \[ 5 \]

This process strips down the complication and shows why understanding these basic rules is so helpful. Simplification is about following these basic steps to make your work easier and more correct.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operations. They form the basic building blocks in algebra. In the exercise you're working on, you dealt with the expression \[ 5 x^0 \]

When simplifying algebraic expressions, such as this one, you use the Zero Exponent Rule to change \( x^0 \) to 1 because it makes managing the terms easier.

Here鈥檚 a quick recap:
  • Recognize the exponent rule that applies, like the Zero Exponent Rule.
  • Apply the rule to simplify the variable part of the expression.
  • Finish by simplifying any remaining parts.
Using these steps can help you tackle various algebraic expressions efficiently.

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Most popular questions from this chapter

In computer science, \(1 \mathrm{KB}\) of memory refers to 1 kilobyte, or 1 \(\times 10^{3}\) bytes, of memory. This is really an approximation of 1 \(\times 2^{10}\) bytes (since computer memory uses powers of \(2)\). The TI- 84 Plus Silver Edition graphing calculator has \(1.5 \mathrm{MB}\) (megabytes) of FLASH ROM, where \(1 \mathrm{MB}\) is \(1000 \mathrm{KB}\). How many bytes of FLASH ROM does this calculator have?

Simplify. $$ \frac{4.2 \times 10^{8}\left[\left(2.5 \times 10^{-5}\right) \div\left(5.0 \times 10^{-9}\right)\right]}{3.0 \times 10^{-12}} $$

For each pair of functions \(f\) and \(g\), determine the domain of \(f / g\) $$ \begin{aligned} &f(x)=3 x-2\\\ &g(x)=2 x-8 \end{aligned} $$

Use the fact that \(10^{3} \approx 2^{10}\) to estimate each of the following powers of \(2 .\) Then compute the power of 2 with a calculator and find the difference between the exact value and the approximation. $$ 2^{31} $$

Simplify. Assume that no denominator is zero and that \(0^{0}\) is not considered. $$ \left(\frac{4 x^{3} y^{5}}{3 z^{7}}\right)^{0} $$
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