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Chapter 7: Problem 71

Simplify each root. $$ \sqrt[3]{x^{3}} $$

Short Answer

Expert verified
The simplified form is \(x\).

Step by step solution

01

Understand the given expression

Given expression is the cube root of \(\root[3]{x^{3}}\). The goal is to simplify this expression.
02

Use the property of roots

According to the property of roots, \(\root[n]{a^{n}} = a\). We can apply this property to our expression.
03

Apply the cube root property

Applying the property \(\root[3]{x^{3}} = x\). Therefore, the simplified form of the given expression is simply \(x\).

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Key Concepts

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

Cube Root Property
To start with, let's discuss the cube root property, which is essential for simplifying cube roots. The cube root property states that for any number or variable, if you have the cube root of that number raised to the third power, the cube root and the exponent cancel each other out.
In mathematical terms, this property is written as: \(\root[n]{a^{n}} = a\).
This means if you take the cube root of a variable like \(x^3\), you're essentially left with \(x\).
Understanding this property helps simplify complex expressions and equations in algebra. It's just like taking out what's inside the root when the inside power matches the root power.
Simplifying Expressions
Simplifying expressions might seem tricky at first, but with practice, it becomes straightforward.
Given the expression \(\root[3]{x^{3}}\), you want to make it as simple as possible by getting rid of unnecessary components.
First, identify the expression inside the root, which in this case is \(x^3\).
Next, apply the cube root property: \(\root[3]{x^3} = x\).
This eliminates the cube and cube root, leaving you with the simplified result of \(x\).
  • Look for exponents that match the root type.
  • Apply the root property.
  • Simplify to the simplest form.
It's like peeling an onion; you remove layers to get to the core. This makes working with more complex algebra easier.
Algebraic Principles
Algebraic principles are key to solving and simplifying expressions.
Two important principles are the power rule and radical rule. The power rule states that \( (a^m)^n = a^{m*n} \), while the radical rule is \( \root[n]{a^m} = a^{\frac{m}{n}} \).
These principles help you manipulate and simplify expressions effectively. Let's use them to understand the original problem better: \( \root[3]{x^3} \).
  • Recognize the base (\

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