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Chapter 1: Problem 60

Write an equivalent expression with negative exponents. $$ \frac{1}{n^{4}} $$

Short Answer

Expert verified
\(n^{-4}\)

Step by step solution

01

- Understand the negative exponent rule

The negative exponent rule states that for any nonzero number 'a' and a positive integer 'm', \(\frac{1}{a^m} = a^{-m}\). This means you can rewrite a fraction as an expression with a negative exponent.
02

- Apply the negative exponent rule

Using the rule from Step 1, recognize that \(n^4\) can be rewritten with a negative exponent when placed in the denominator. \(\frac{1}{n^4}\) becomes \(n^{-4}\).

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Key Concepts

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

Negative Exponent Rule
Understanding the negative exponent rule helps simplify expressions. The rule states: for any nonzero number 'a' and a positive integer 'm', we have \(\frac{1}{a^m} = a^{-m}\). This means when you see a fraction like \(\frac{1}{a^m}\), you can rewrite it as an expression with a negative exponent, \a^{-m}\.
Imagine we have \(\frac{1}{n^4}\). According to the negative exponent rule, this can be expressed as \^{-4}\. Use this rule to transform any fractions where the variable or number is in the denominator with a positive exponent. It helps simplify calculations and makes it easier to work with expressions in equations.
Equivalent Expressions
An equivalent expression is a different way of writing the same mathematical expression. By using the negative exponent rule, we can convert fractions into expressions with negative exponents, making them equivalent.
For example, converting \(\frac{1}{n^4}\) into \^{-4}\ shows that both expressions hold the same value even though they look different.
To find equivalent expressions, recognize patterns and apply properties of exponents. Converting between forms allows for flexibility in solving different kinds of problems. It's a useful skill in algebra and helps in simplifying complex expressions.
Properties of Exponents
Exponents have specific properties that simplify calculations and solve equations. These properties are rules that apply to numbers with exponents which can be formulated for easier manipulation.
Some key properties of exponents are:
  • Product of Powers: \(a^m \times a^n = a^{m+n}\)
  • Quotient of Powers: \(a^m / a^n = a^{m-n}\)
  • Power of a Power: \((a^m)^n = a^{m \times n}\)
  • Zero Exponent: \(a^0 = 1\) for any nonzero 'a'
  • Negative Exponent: \(a^{-m} = \frac{1}{a^m}\)

Using these properties, you can convert and simplify expressions efficiently. For example, recognizing \( \frac{1}{n^4} \) and converting it to \ n^{-4} \ gains the benefits of these properties in algebraic manipulation. Understanding and applying these rules make working with exponents straightforward and less prone to errors.

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Most popular questions from this chapter

Find an equivalent expression by factoring. $$ x y+x $$

Write an equivalent expression using the distributive law. $$ 7(x+1) $$

To the student and the instructor: Synthesis exercises are designed to challenge students to extend the concepts or skills studied in each section. Many synthesis exercises require the assimilation of skills and concepts from several sections. On a quiz, Mia answers \(6 \in \mathbb{Z}\) while Giovanni writes \(\\{6\\} \in \mathbb{Z} .\) Giovanni's answer does not receive full credit while Mia's does. Why?

Write an equivalent expression using the distributive law. $$ 5 x(y-z+w) $$

Which numbers in the list provided are (a) whole numbers? (b) integers? (c) rational numbers? (d) irrational numbers? (e) real numbers?. $$ -8.7,-3,0, \frac{2}{3}, \sqrt{7}, 6 $$
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