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Chapter 1: Problem 22

Convert to expressions with rational exponents. \(\sqrt[8]{c}\)

Short Answer

Expert verified
The expression is \(c^{\frac{1}{8}}\).

Step by step solution

01

Understand the problem

Given the radical expression \(\root[8]{c}\), it needs to be converted into an expression with rational exponents.
02

Identify the conversion rule

The conversion from radical form to rational exponents uses the rule \(\root[n]{a} = a^{\frac{1}{n}}\).
03

Apply the rule

Using the rule \(\root[n]{a} = a^{\frac{1}{n}}\), with \(n = 8\) and \(a = c\), convert \(\root[8]{c}\) to exponential form: \(c^{\frac{1}{8}}\).

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Key Concepts

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

Radical Expressions
Radical expressions involve roots, such as square roots, cube roots, and in our case, an eighth root. The general form of a radical expression is \(\root[n]{a}\), where \(a\) is the radicand (the number or expression inside the root), and \(n\) is the index (the root we are taking). For example, \(\root[8]{c}\) means the eighth root of \(c\). Radicals can be tricky to work with because they involve understanding how roots operate and combine with other numbers. However, converting them into exponential form, which uses rational exponents, can make calculations much simpler.
Exponential Form
Exponential form expresses numbers as powers or exponents. In this form, we use \(a^b\), where \(a\) is the base and \(b\) is the exponent. This includes rational exponents, which are fractions. Rational exponents let us write roots as exponents. For example, instead of writing \(\root[n]{a}\), we can write \(a^{\frac{1}{n}}\). This conversion helps to simplify and solve more complex mathematical problems. For instance, in our exercise, we converted the radical expression \(\root[8]{c}\) into the exponential form \(c^{\frac{1}{8}}\). Both are equivalent, but exponential form can be easier to manipulate.
Conversion Rule
To convert from a radical expression to an expression with rational exponents, we use a simple rule: \(\root[n]{a} = a^{\frac{1}{n}}\). This rule allows us to change any radical into an exponent form. Let's break this rule down:
  • Identify the radicand (the value inside the root) as \(a\).
  • Identify the index of the root as \(n\).
  • Rewrite the expression with the base \(a\), and the exponent as \(\frac{1}{n}\).
Using our example from the exercise, \(\root[8]{c}\) becomes \(\root[8]{c} = c^{\frac{1}{8}}\). Remembering and applying this rule can make working with roots much easier, transforming them into a more straightforward exponential form that can simplify calculations and further algebraic manipulations.

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Most popular questions from this chapter

Use a calculator to find the radian measure of an acute angle whose trigonometric function is given. \(\tan t=1.26\)

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