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Chapter 8: Problem 51

Rewrite the radical expression in exponential notation. $$\sqrt[99]{x}$$

Short Answer

Expert verified
x^{\frac{1}{99}}

Step by step solution

01

Identify the Radical Expression

Recognize the given radical expression: \(\textrm{\textbackslash sqrt[99]\textbraceleft x\textbraceright}\). This is a 99th root.
02

Understand the Radical to Exponential Conversion

The general rule to convert a radical expression \(\textrm{\textbackslash sqrt[n]\textbraceleft x\textbraceright}\) to exponential form is: \(\textrm{x\textasciicircum \textbraceleft \frac\textbraceleft 1\textbraceright \textbraceleft n\textbraceright \textbraceright}\).
03

Apply the Conversion Rule

Using the rule, convert \(\textrm{\textbackslash sqrt[99]\textbraceleft x\textbraceright}\) to exponential notation: \(\textrm{x\textasciicircum \textbraceleft \frac\textbraceleft 1\textbraceright \textbraceleft 99\textbraceright \textbraceright}\).

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Key Concepts

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

Understanding Radical Expression
Radical expressions often look complicated, but they can be simplified once you know the basics. A radical expression involves roots, which are the opposite of exponents. The square root, cube root, and in our case, the 99th root, are all examples of radical expressions.

A radical expression like \( \sqrt[99]{x} \) asks the question: 'What number, when raised to the power of 99, gives us x?' The symbol \( \sqrt{} \) is called a radical, and the number inside it is called the radicand. The small number outside, 99 in this case, is the index.

It's important to get comfortable with these terms because they'll make understanding later topics easier.
Switching to Exponential Notation
Exponential notation is a way to express numbers and variables using exponents. It often simplifies calculations and helps to understand the behavior of functions.

The fundamental idea is that a radical expression can be converted into exponential form using the general rule: \( \sqrt[n]{x} = x^{1/n} \).

This conversion makes it easier to manipulate and solve equations because it removes the root and replaces it with an exponent. For example, \( \sqrt[99]{x} \) becomes \( x^{1/99} \). This form is more convenient for many algebraic operations and calculus.
Mastering Algebraic Conversion
The algebraic conversion between radical expressions and exponential notation is a powerful tool in algebra. Here鈥檚 why:
  • It simplifies many types of calculations, especially when dealing with higher roots.
  • It helps in comparing the sizes of numbers and expressions.
  • In calculus, it makes differentiation and integration more straightforward.

To convert a radical expression to exponential notation, always remember this rule of thumb: the root's index becomes the denominator of a fraction.

For the given example, \( \sqrt[99]{x} \), the index, 99, goes into the denominator: \( x^{1/99} \). Practicing this conversion will eventually make it second nature and significantly improve your algebra skills.

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Most popular questions from this chapter

Simplify. $$\sqrt[4]{8} \sqrt[4]{6}$$

Simplify. $$\left(y^{\frac{1}{6}}\right)^{\frac{1}{3}}$$

Rewrite the radical expression in exponential notation and simplify. $$\sqrt[20]{a^{5}}$$

The completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Simplify: \(\sqrt{\frac{100}{c}}\)

Simplify. $$\frac{x^{\frac{4}{5}}}{x^{\frac{1}{5}}}$$
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