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Chapter 5: Problem 13

Use the zero and negative exponent rules to simplify each expression. \(3^{0}\)

Short Answer

Expert verified
The simplification of the expression \(3^{0}\) is 1.

Step by step solution

01

Recall the Zero Exponent Rule

Remember the rule that any non-zero number raised to the power of 0 equals 1. The zero exponent rule states that for any non-zero real number \(a\): \(a^{0} = 1\).
02

Use the Rule to the Expression

Applying the zero exponent rule to the given expression \(3^{0}\), we can replace \(3^{0}\) with 1.

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Key Concepts

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

Exponent Rules
Exponent rules are essential for managing powers and simplifying expressions involving exponents. These rules help maintain consistency and simplify calculations. Here are the main exponent rules you should know:
  • Zero Exponent Rule: Any non-zero number raised to the power of zero is equal to one, e.g., \(a^{0} = 1\).
  • Product of Powers Rule: When multiplying two exponents with the same base, add the exponents: \(a^{m} \times a^{n} = a^{m+n}\).
  • Quotient of Powers Rule: When dividing two exponents with the same base, subtract the exponents: \(a^{m} \div a^{n} = a^{m-n}\).
  • Power of a Power Rule: When raising an exponent to another power, multiply the exponents: \((a^{m})^{n} = a^{mn}\).
These rules allow you to manipulate and solve expressions more easily. Remembering these basics will make working with exponents much simpler.
Simplifying Expressions
Simplifying expressions involves reducing them to their simplest form while maintaining the same value. This process often involves using the exponent rules. For example, consider the expression \(3^{0}\). Knowing the zero exponent rule allows us to simplify it directly to 1.
Simplification helps in:
  • Making mathematical expressions easier to work with and understand.
  • Decreasing the complexity of larger problems by focusing on smaller, manageable parts.
  • Solving problems more accurately by providing a clear and concise representation of the expression.
Using the appropriate rules for exponents can significantly streamline your problem-solving process, making it more efficient.
Negative Exponents
Negative exponents might seem tricky at first, but they have a straightforward interpretation. A negative exponent indicates that the base should be reciprocated. For instance, \(a^{-n} = \frac{1}{a^{n}}\). This simplifies the way you can manage expressions with negative exponents.
  • Flip the base to its reciprocal.
  • Convert the negative exponent to a positive exponent.
If you are presented with an expression like \(2^{-3}\), according to the rule, it equates to \(\frac{1}{2^{3}}\) or \(\frac{1}{8}\). Understanding negative exponents allows deeper manipulation and solution of complex mathematical expressions.
Mathematical Problem-Solving
Mathematical problem-solving often requires a combination of understanding the concepts, recognizing patterns, and applying rules. The key is breaking down the problem into simpler parts. For cases involving exponents, knowledge of the rules discussed is crucial.
  • Identify the type of exponent in the expression: positive, negative, or zero.
  • Choose the appropriate rule or formula to apply.
  • Simplify step-by-step and check your work as you go.
In our example of \(3^{0}\), identifying it as zero power leads to immediate simplification using the zero exponent rule. Problem-solving involves practice, and by repeatedly applying these rules, you will improve your ability to simplify and solve various mathematical problems efficiently.

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Most popular questions from this chapter

You are offered a job that pays $$\$ 30,000$$ for the first year with an annual increase of \(5 \%\) per year beginning in the second year. That is, beginning in year 2 , your salary will be \(1.05\) times what it was in the previous year. What can you expect to earn in your sixth year on the job? Round to the nearest dollar.

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=-\frac{1}{16}, r=-4\)

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=4, r=2\)

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(15,30,60,120, \ldots\)

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=5000, r=1\)
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