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Chapter 5: Problem 35

Write each of the following in radical form. For example, \(3 x^{\frac{2}{3}}=3 \sqrt[3]{x^{2}}\). \((2 y)^{\frac{1}{3}}\)

Short Answer

Expert verified
The radical form of \\( (2y)^{\frac{1}{3}} \\\) is \\(\sqrt[3]{2y} \\\).

Step by step solution

01

Understanding the Problem

The expression given is \( (2y)^{\frac{1}{3}} \), which is written in exponential form. The task is to convert this into radical form.
02

Identifying the Base and Exponent

In the expression \( (2y)^{\frac{1}{3}} \), \(2y\) is the base and \(\frac{1}{3}\) is the exponent. In fractional exponents, the numerator represents the power, and the denominator represents the root.
03

Applying the Radical Form Formula

To convert \( (2y)^{\frac{m}{n}} \) to radical form, it becomes \(\sqrt[n]{(2y)^m} \). Here, \m\ is 1 and \ is 3, leading to \(\sqrt[3]{(2y)} \).
04

Writing in Radical Form

For the expression \( (2y)^{\frac{1}{3}} \), use the formula to rewrite it as \(\sqrt[3]{2y} \), indicating the cube root of \2y\.

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Key Concepts

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

Exponential Form
Exponential form is a mathematical expression that represents repeated multiplication of the same number. It is written as a base raised to an exponent, like \( (2y)^{\frac{1}{3}} \). Here, the base is \(2y\), and the exponent is \(\frac{1}{3}\). This format is concise and powerful because it allows us to easily manipulate and understand expressions involving powers.
  • **Base**: This is the number that is multiplied by itself. In our case, the base is \(2y\).
  • **Exponent**: This indicates how many times the base is used in the multiplication. Here, the exponent is \(\frac{1}{3}\), which we'll explore further as it's a fractional exponent.
In exponential form, understanding the relationship between base and exponent equips you with the ability to switch between powerful expressions and their radical forms. This interchangeability is especially useful in algebra and calculus.
Fractional Exponents
Fractional exponents are a way to express powers and roots together using fractions. They are quite versatile, making complex expressions manageable. Let's break down their meaning using \((2y)^{\frac{1}{3}}\) as an example.
  • **Numerator**: The top part of the fraction represents the usual power or exponent. For \((2y)^{\frac{1}{3}}\), the numerator is 1, meaning the base \(2y\) is simply raised to the first power. So, it's not added complexity in terms of "powering up."
  • **Denominator**: The bottom of the fraction specifies the root to be taken. Here, it is 3, indicating a cube root.
When you see a fractional exponent, it's simply telling you to perform a root operation. They're not as intimidating once you know that the numerator is a power and the denominator is a root. Converting between fractional exponents and radicals becomes a straightforward process as you practice.
Cube Root
A cube root is a special type of root in which you are looking for a number that multiplies by itself three times to give the original number. The cube root of a number \(x\) is denoted as \(\sqrt[3]{x}\). Let's apply this to our expression.In \(\sqrt[3]{2y}\), you are finding the number that, when multiplied by itself three times, equals \(2y\). This process is the reverse of cubing a number.
  • **Understanding**: Imagine having cubes of side length equivalent to the base \(2y\) and you're determining the side length that will produce the original volume after multiplying it by itself twice more.
  • **Application**: Practically, cube roots are used in scenarios involving three-dimensional geometric calculations, physical applications like volume, and simplifying expressions in algebra.
Practicing cube roots in different expressions not only helps in understanding radicals but also in solving polynomial equations and understanding complex geometric applications. Remember, the key is looking for what when cubed returns to the original base.

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Most popular questions from this chapter

Simplify each of the following. Express final results using positive exponents only. For example,\(\left(2 x^{\frac{1}{2}}\right)\left(3 x^{\frac{1}{3}}\right)=6 x^{\frac{5}{6}}\). \(\left(\frac{2 x^{\frac{1}{3}}}{3 y^{\frac{1}{4}}}\right)^{4}\)

Perform the indicated operations and express answers in simplest radical form. (See Example 5.) \(\frac{\sqrt[3]{3}}{\sqrt[4]{3}}\)

Use scientific notation and the properties of exponents to help you perform the following operations. \(\sqrt[3]{0.001}\)

Perform the indicated operations and express answers in simplest radical form. (See Example 5.) \(\frac{\sqrt{9}}{\sqrt[3]{3}}\)

For Problems \(19-32\), write each of the following in ordinary decimal notation. For example, \((3.18)(10)^{2}=318\). \((1.62)(10)^{2}\)
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