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Chapter 10: Problem 8

$$\text { Fill in the blanks. } n^{5} \cdot n=n \square^{6}+\square=n^{6}$$

Short Answer

Expert verified
The blanks should be filled with `n` and `0` respectively.

Step by step solution

01

Apply Exponent Rules

According to the laws of exponents, when you multiply two expressions with the same base, you add their exponents. Here, the base is `n` for both terms. Therefore, we add the exponents 5 and 1: \[n^{5} \times n = n^{5+1}\]
02

Simplify the Expression

Simplify the exponent addition from the previous step: \[n^{5+1} = n^{6}\]
03

Complete the Equation

Replace the squares in the original problem with the values obtained: \[n^{5} \times n = n\square^{6} + \square = n^{6}\] Thus, the blanks should be filled as follows to make the equation true: \[n^{5} \times n = n^{6}+0 = n^{6}\]

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Key Concepts

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

headline of the respective core concept
When working with exponents, multiplication is one of the fundamental operations. The key rule to remember is that when you multiply two numbers with the same base, you add their exponents. For example, consider the expression: \[n^5 \times n^2\]Here, the base is \(n\), and the exponents are 5 and 2. Using the multiplication of exponents rule, we add the exponents: \[n^{5+2} = n^7\]This simplifies the multiplication of exponents process. Remember, the bases must be the same for this rule to apply.
headline of the respective core concept
Simplifying exponents involves reducing expressions to their simplest form by applying the rules of exponents. In the given exercise, the term \(n^5\times n\) simplifies because it follows the exponent rules. Breaking it down step-by-step:
  • Identify the bases and exponents. Here, the base is \(n\) for both terms, and the exponents are 5 and 1.
  • Apply the rule of exponents for multiplication: Add the exponents.
  • This gives us: \(n^5 \times n = n^{5+1} = n^6\).
This simplification process helps convert complex expressions into more manageable forms.
headline of the respective core concept
Understanding exponent rules for the same base is crucial when simplifying expressions involving exponents. Here are the primary rules to keep in mind:
  • Multiplication Rule: For the same base, add the exponents. \[a^m \cdot a^n = a^{m+n}\]
  • Division Rule: For the same base, subtract the exponents. \[\frac{a^m}{a^n} = a^{m-n}\]
  • Power of a Power Rule: Multiply the exponents. \[(a^m)^n = a^{mn}\]
Specifically, the exercise provided uses the multiplication rule, reinforcing the concept that with the same base, the exponents are added, simplifying the original expression.

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Most popular questions from this chapter

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