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Chapter 4: Problem 24

\(6^{0}-13^{0}\)

Short Answer

Expert verified
The result is 0.

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 one. This means for any number 'a', if a^0, then it is always equal to 1 as long as 'a' is not zero.
02

Apply the Zero Exponent Rule

Apply the zero exponent rule to both parts of the expression. For the first part, we have 6^0, and according to the zero exponent rule, 6^0 = 1. For the second part, we have 13^0, and similarly, 13^0 = 1.
03

Perform the Subtraction

Subtract the values obtained in Step 2. Since both 6^0 and 13^0 equal 1, the subtraction is 1 - 1.
04

Simplify the Result

Simplify the result from the subtraction. Therefore, 1 - 1 = 0.

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Key Concepts

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

Zero Exponent Rule Explained
The zero exponent rule is a fundamental concept in algebra. It states that any non-zero number raised to the power of zero is always one. This can be written mathematically as: \[ a^0 = 1, \text{ for any non-zero number } a \]. For example, in our exercise, we had to simplify \( 6^0 - 13^0 \). Using the zero exponent rule, we know that both \( 6^0 \) and \( 13^0 \) equal 1. So, the problem becomes \( 1 - 1 \). Understanding this rule helps simplify many algebraic expressions and is useful for solving more complex equations.
Understanding Exponents
Exponents are a way to express repeated multiplication of a number by itself. For instance, \( 3^4 \) means multiplying 3 by itself four times: \[ 3^4 = 3 \times 3 \times 3 \times 3 = 81 \]. Exponents have specific properties, including the zero exponent rule and others like the product rule \( a^m \cdot a^n = a^{m+n} \), and the power rule \( (a^m)^n = a^{mn} \). Mastering these properties is essential for simplifying and solving algebraic expressions.
Basic Algebra and Simplification
Basic algebra involves combining like terms and using rules of arithmetic to simplify expressions. Simplification often involves:
  • Applying the zero exponent rule
  • Combining like terms
  • Using arithmetic operations like addition, subtraction, multiplication, and division
In our exercise, we applied the zero exponent rule to transform \( 6^0 - 13^0 \) into \( 1 - 1 \). Then through basic subtraction, we simplified \( 1 - 1 \) to get the final result of 0. Simplification is crucial for solving and understanding more complex algebraic problems. Each step breaks the problem into smaller, manageable pieces, making it easier to arrive at the correct solution.

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Most popular questions from this chapter

$$\text { Subtract the sum of } 9 t^{3}-3 t+8 \text { and } t^{2}-8 t+4 \text { from the sum of } 12 t+8 \text { and } t^{2}-10 t+3$$

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