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Chapter 8: Problem 40

Rewrite the expression with positive exponents. $$(-10 a)^{0}$$

Short Answer

Expert verified
The simplified expression is 1, provided that \(a\) is not zero. If \(a\) is zero, the expression is undefined.

Step by step solution

01

Analyze the base and exponent

Here we have an expression with a base of \(-10a\) and an exponent of 0. The base is not 0 (unless \(a\) is equal to 0), so the expression is well defined.
02

Use the rule for zero exponent

The rule of exponents states that any non-zero number raised to the power of zero is equal to 1. So, we simplify \((-10a)^{0}\) to 1, provided that \(a\) is not zero.
03

Write the simplified expression

Following the rule applied in Step 2, the simplified expression is 1, as long as \(a\) is not zero. If \(a\) is zero, then the expression is undefined because we have a 0 base to a 0 power.

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Key Concepts

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

Understanding the Zero Exponent Rule
When dealing with exponents, one of the most important rules to remember is the zero exponent rule. This rule states that any non-zero number raised to the power of zero is equal to 1.
For example, whether it's a simple number like 5 or a more complex expression like \((-10a)\), if it's raised to the 0 power, the result is 1. This is true as long as the base itself isn't zero, because 0 raised to the 0 is undefined.
  • This rule helps in simplifying expressions powerfully and quickly.
  • The zero exponent rule applies to variables and more complex bases.
Whenever you see an expression with an exponent of 0, remember, the answer is simply 1, assuming the base isn't zero.
Simplifying Expressions with Exponents
Simplifying expressions often involves dealing with exponents. Knowing how to handle them correctly can make your work much simpler.
For example, simplifying an expression with a zero exponent is straightforward thanks to the zero exponent rule.
  • Start by identifying the base and exponent.
  • Check if the exponent is zero.
  • Apply any known exponent rules like the zero exponent rule.
Through this process, something like \((-10a)^{0}\) becomes just 1, as long as a isn't zero.
With practice, simplifying such expressions becomes second nature. You'll find that once you grasp these rules, many algebraic expressions are easier to manage.
Base and Exponent Analysis
Understanding the components of an expression is crucial. Every power expression has two main parts: the base and the exponent.
The base is the number or expression being multiplied, and the exponent tells you how many times to multiply that base by itself.
  • In \((-10a)^{0}\), the base is \(-10a\).
  • The exponent is 0, indicating application of the zero exponent rule.
Let's remember that when analyzing
  • The base must be non-zero for the zero exponent rule to apply.
  • Understanding the base helps reveal what happens to the overall expression under different exponent rules.
By recognizing the base and exponent clearly, you can quickly apply the right rules and simplify expressions without confusion.

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