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Chapter 8: Problem 112

Simplify each radical. $$ \sqrt[3]{y^{18}} $$

Short Answer

Expert verified
The simplified form is \ y^6 \.

Step by step solution

01

- Understand the problem

We need to simplify the radical expression \( \sqrt[3]{y^{18}} \). This means we are looking for the simplest form of the expression under a cube root.
02

- Apply the property of exponents

Use the property of exponents that states \[ \sqrt[3]{y^n} = y^{n/3} \], where \ n = 18 \ in this case. This property allows us to simplify radicals involving exponents.
03

- Simplify the exponent inside the radical

Substitute \ n = 18 \ into the expression \[ \sqrt[3]{y^{18}} = y^{18/3} = y^6 \].

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Key Concepts

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

Cube Root
A cube root is a special type of root used to find a number that, when multiplied by itself three times, results in the original number. The cube root symbol looks like \( \sqrt[3]{\cdot} \). Unlike square roots, cube roots can return both positive and negative values.
In our exercise, we start with \( \sqrt[3]{y^{18}} \). To simplify this, we need to understand how cube roots interact with exponents.
Exponent Rules
Exponent rules are essential for simplifying expressions involving radicals. The key rule we use here is that \[ \sqrt[3]{y^n} = y^{n/3} \]. This property allows us to break down complex radical expressions into simpler forms.
In the example, \( n \) is 18, so we substitute it into the rule: \[ \sqrt[3]{y^{18}} = y^{18/3} = y^6 \].
This shows how exponents inside radicals can be simplified for easier calculations.
Radical Expressions
Radical expressions involve roots, such as square roots or cube roots. Simplifying them often involves breaking down the expression inside the radical to its simplest form.
In our case, we start with a cube root, \( \sqrt[3]{y^{18}} \). Using the exponent rule, we simplify it to \( y^6 \).
By understanding how to manipulate both radicals and exponents, we make complex expressions much simpler to work with. This mastery helps in various areas of mathematics, including algebra and calculus.

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Most popular questions from this chapter

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