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Chapter 5: Problem 35

Express using negative exponents. $$x^{5}$$

Short Answer

Expert verified
\(x^{5} = \frac{1}{x^{-5}}\)

Step by step solution

01

Identify the given expression

The given expression is \(x^{5}\).
02

Understanding Exponents

Recall that any positive exponent \(n\) can be expressed with a negative exponent by using the reciprocal, e.g., \(a^n = \frac{1}{a^{-n}}\).
03

Convert to Negative Exponents

Now, express \(x^{5}\) using a negative exponent, which would be: \(\frac{1}{x^{-5}}\).

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Key Concepts

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

mathematical expressions
When dealing with mathematical expressions, it's important to understand what exponents are and how they function. An exponent (also known as a power) refers to a number that tells us how many times a base number is multiplied by itself. In the expression \(x^5\), \(x\) is the base and \(5\) is the exponent. This means \(x\) is multiplied by itself 5 times: \(x \cdot x \cdot x \cdot x \cdot x\). Understanding this basic concept is essential before diving into more complex ideas like negative exponents.
exponent rules
Exponent rules help us simplify and manipulate expressions involving exponents. Let's explore some key rules:
  • Product of Powers Rule: When multiplying two exponents with the same base, add their powers. For example, \(x^a \cdot x^b = x^{a+b}\).
  • Power of a Power Rule: When raising an exponent to another exponent, multiply their powers. For example, \((x^a)^b = x^{a \cdot b}\).
  • Negative Exponent Rule: A negative exponent means the reciprocal of the base raised to the opposite positive exponent. For example, \(a^{-n} = \frac{1}{a^n}\). This is the rule we use when converting \(x^5\) into a form involving a negative exponent. Following this rule, \(x^5\) can be written as \(\frac{1}{x^{-5}}\).
Practicing these rules will help you simplify and understand complex mathematical expressions more easily.
reciprocals
Reciprocals play an important role in understanding negative exponents. The reciprocal of a number is 1 divided by that number. For instance, the reciprocal of 4 is \(\frac{1}{4}\). In the context of exponents, if you have \(x^{-n}\), it signifies the reciprocal of \(x^n\). Learning how reciprocals work can help you grasp why \(x^5\) becomes \(\frac{1}{x^{-5}}\) when expressed with a negative exponent. In essence, a negative exponent flips the fraction, emphasizing the inverse relationship between the base and its power. Understanding reciprocals solidifies your comprehension of mathematical expressions involving exponents.

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Most popular questions from this chapter

Simplify. Assume that no denominator is zero and that \(0^{0}\) is not considered. $$ \left(\frac{-4 p^{8}}{3 m^{2} n^{3}}\right)^{3} $$

For each pair of functions fand \(g\), determine the domain of the sum, the difference, and the product of the two functions. $$ \begin{aligned} &f(x)=x^{2}\\\ &g(x)=7 x-4 \end{aligned} $$

For each pair of functions fand \(g\), determine the domain of the sum, the difference, and the product of the two functions. $$ \begin{aligned} &f(x)=\frac{1}{x-3}\\\ &g(x)=4 x^{3} \end{aligned} $$

For each pair of functions fand \(g\), determine the domain of the sum, the difference, and the product of the two functions. $$ \begin{aligned} &f(x)=5 x-1\\\ &g(x)=2 x^{2} \end{aligned} $$

Simplify. $$ \frac{\left(\frac{1}{2}\right)^{3}\left(\frac{2}{3}\right)^{4}}{\left(\frac{5}{6}\right)^{3}} $$
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