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Chapter 10: Problem 382

In the following exercises, simplify. $$\frac{n^{-4}}{n^{-10}}$$

Short Answer

Expert verified
\tex\(n^6\tex\)

Step by step solution

01

Write the expression in a single fraction

Start with the given expression \(\frac{n^{-4}}{n^{-10}}\tex\) and recognize that the rule for dividing powers of the same base is: \tex\(a^m / a^n = a^{m-n}\tex\).
02

Apply the exponent division rule

Apply the rule by subtracting the exponent in the denominator from the exponent in the numerator: \tex\(n^{-4 - (-10)} = n^{-4 + 10}\tex\).
03

Simplify the exponent

Simplify the expression by computing \tex\(-4 + 10\tex\), which equals \tex\(6\tex\). Thus, \tex\(n^{-4 + 10} = n^6\tex\).

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Key Concepts

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

Exponent Rules
Understanding exponent rules is fundamental when simplifying expressions involving exponents.
The key rules to keep in mind are:
  • Product of Powers: When multiplying two exponents with the same base, you add their powers, i.e., \(a^m \times a^n = a^{m+n}\).
  • Quotient of Powers: When dividing two exponents with the same base, you subtract the exponent of the denominator from the exponent of the numerator, i.e., \(a^m / a^n = a^{m-n}\).
  • Power of a Power: When raising an exponent to another power, you multiply the exponents, i.e., \((a^m)^n = a^{m \times n}\).
By mastering these rules, you can efficiently simplify even complex expressions involving exponents.
Dividing Exponents
Dividing exponents is a common operation in algebra and it's important to understand how to handle it properly.
When you come across an expression like \(\frac{n^{-4}}{n^{-10}}\), you need to apply the Quotient of Powers rule mentioned earlier.
Let's break it down:
  • Identify the base: In \(\frac{n^{-4}}{n^{-10}}\), the base is \(n\). Both the numerator and the denominator have this same base.
  • Subtract the exponents: According to the Quotient of Powers rule, subtract the exponent in the denominator from the exponent in the numerator: \(-4 - (-10)\).
  • Simplify the subtraction: Remember that subtracting a negative is the same as adding: \(-4 + 10 = 6\).
So, \(\frac{n^{-4}}{n^{-10}} = n^{6}\).
Simplifying Expressions
Simplifying expressions involves reducing them to their most basic form.
When dealing with exponents, this often means applying exponent rules.
  • Combine like terms: Use exponent rules to combine terms with the same base.
  • Perform arithmetic operations: Carry out any necessary arithmetic, such as addition, subtraction, multiplication, or division, on the exponents.

Let's recap our example: \(\frac{n^{-4}}{n^{-10}}\).
By subtracting the exponents (following the Quotient of Powers rule), we simplified the expression to \(n^{6}\).
This expression, \(n^{6}\), is the simplest form and cannot be simplified further unless additional context or operations are given.
Remember, breaking down complex expressions step-by-step and consistently applying the rules will make the process of simplifying much easier.

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