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Chapter 5: Problem 105

Simplify. Do not use negative exponents in the answer. \(15(-6 x)^{0}\)

Short Answer

Expert verified
The expression simplifies to 15.

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. Thus, \[(-6x)^0 = 1\] regardless of the value of \(x\).
02

Simplify using the Zero Exponent Rule

Apply the zero exponent rule to the expression. This makes the expression:\[15 imes 1 = 15\] This is because \((-6x)^0 = 1\).
03

Final Simplified Expression

After applying the zero exponent rule and performing the multiplication, the simplified expression is:\[15\]

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Key Concepts

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

Simplifying Expressions
When simplifying expressions, your main goal is to make them as easy and clear as possible. In mathematics, this often involves applying specific rules or eliminating unnecessary parts. For the expression \(15(-6x)^0\), the task is to simplify it by applying known exponent rules. Once the zero exponent rule is recognized, the expression becomes less complicated. This rule allows us to reduce
  • unnecessary variables
  • complex terms
  • repetitions
By following these steps, simplifying ensures that students can work with formulas in a straightforward and logical way. As you become comfortable with these concepts, you'll find that simplifying expressions gets easier and more instinctive.
Negative Exponents
Negative exponents can be confusing, but they're important to understand. They indicate the reciprocal of the base raised to the corresponding positive exponent. Here's how it works:
  • A base raised to a negative exponent is equal to one divided by the base raised to the positive exponent.
  • For example, \(x^{-n} = \frac{1}{x^n}\).
  • This rule helps in transforming negative exponents to positive ones, which are easier to manage.
While the original exercise doesn't involve negative exponents directly, understanding this principle is crucial. If we faced an expression like \(-6x^{-1}\), knowing the negative exponent rule would let us rewrite it as \(-\frac{6}{x}\). Remember, the goal is to have positive exponents in your final answer to keep expressions clean and manageable.
Exponent Rules
Exponent rules are a fundamental part of simplifying expressions. These rules provide a framework that makes working with powers easier. The original problem highlights two key rules:
  • **Product Rule**: This states that when multiplying like bases, you should add their exponents (\(a^m \cdot a^n = a^{m+n}\)).

  • **Zero Exponent Rule**: Any non-zero number raised to the power of zero equals 1 (i.e., \(a^0 = 1\)). In the problem, \((-6x)^0 = 1\), simplifying the rest of the expression easily.

  • **Quotient Rule**: When dividing like bases, subtract the exponents (\(a^m / a^n = a^{m-n}\)). This rule wasn't needed for the given problem but is useful in other contexts.
These rules help you tackle diverse problems, whether it involves simplifying a single expression or solving complex equations. They provide consistency and ensure that any calculations made are mathematically sound. Once you internalize these rules, manipulating and simplifying expressions becomes a more intuitive process.

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Most popular questions from this chapter

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