Problem 366 In the following exercises, simp... [FREE SOLUTION]

91影视

Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

$Math Studyset 91影视 Explanations$ Math

2016 Edition

Chapter 10: Problem 366

In the following exercises, simplify. $$\left(n^{2}\right)^{-1}$$

Short Answer

Expert verified

\[^{-2} = \frac{1}{n^{2}} \]

Step by step solution

Identify the Exponent Rule

Recognize that the expression inside the parentheses, $^{2}$, is raised to the power of $-1$. The exponent rule to apply here is \[ \left(a^{m}\right)^{n} = a^{mn} .\]

Apply the Exponent Rule

Use the rule identified in Step 1. Multiply the exponents: $(n^{2})^{-1} = n^{2 \cdot -1}$

Simplify the Exponent

Multiply the exponents: \[ \left(n^{2}\right)^{-1} = n^{-2}\]

Use the Negative Exponent Rule

A negative exponent means we take the reciprocal of the base: \[ n^{-2} = \frac{1}{n^{2}} \]

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Key Concepts

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

Exponent Rules

Exponent rules help us simplify expressions involving exponents. They are important when performing calculations that involve power terms. There are various rules we need to understand:

Product Rule: When multiplying two powers with the same base, add the exponents. For example: \[ a^m \times a^n = a^{m+n} \]
Quotient Rule: When dividing two powers with the same base, subtract the exponent of the denominator from the exponent of the numerator. For example: \[ \frac{a^m}{a^n} = a^{m-n} \]
Power Rule: When raising a power to another power, multiply the exponents. For example: \[ \left(a^m\right)^n = a^{m \times n} \]

These rules are fundamental when working with exponents in algebra. They allow us to rewrite complicated expressions in simpler forms, making it easier to solve problems. Remember, practice is key to mastering them.

Parentheses in Exponents

Parentheses in exponent expressions indicate that everything inside the parentheses is affected by the exponent outside. This is especially useful when we apply the Power Rule. For example, given the expression $(n^{2})^{-1}$:

First, understand that $ n^2 $ is inside the parentheses.
Next, the whole expression $ n^2 $ is raised to the power of $ -1 $.
Using the Power Rule, we multiply the exponents: $ \left(n^2\right)^{-1} = n^{2 \times -1} $.

This simplifies to $ n^{-2} $, significantly transforming the original expression to a simpler form. Understanding the impact of parentheses is crucial, as it ensures we apply the correct operations in the right order.

Negative Exponent Rule

The negative exponent rule helps us deal with negative powers. When you see a negative exponent, it indicates you should take the reciprocal of the base raised to the corresponding positive exponent. For example:

Given the expression $ n^{-2} $:
Know that $ n^{-2} $ means $ 1 $ divided by $ n^{2} $.
Thus, $ n^{-2} = \frac{1}{n^{2}} $.

This rule transforms a potentially complicated negative exponent into a manageable fraction. Understanding this rule is especially important for dealing with expressions with negative exponents, simplifying and ensuring you can solve equations efficiently.

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91影视

In the following exercises, simplify. $$\left(n^{2}\right)^{-1}$$

Short Answer

Step by step solution

Identify the Exponent Rule

Apply the Exponent Rule

Simplify the Exponent

Use the Negative Exponent Rule

Key Concepts

Exponent Rules

Parentheses in Exponents

Negative Exponent Rule

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