Simplify The Expression: $\[ \sqrt{21} \left( \sqrt[3]{162} + \sqrt[3]{64} \right) \\]

Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a deep understanding of various mathematical concepts, including exponents, radicals, and algebraic manipulations. In this article, we will focus on simplifying a specific expression involving radicals and exponents. The given expression is ${ \sqrt{21} \left( \sqrt[3]{162} + \sqrt[3]{64} \right) }$. Our goal is to simplify this expression by applying various mathematical techniques and formulas.

Understanding the Expression

Before we start simplifying the expression, let's break it down and understand its components. The expression consists of two main parts: $\sqrt{21}$ and $\left( \sqrt[3]{162} + \sqrt[3]{64} \right)$. The first part involves the square root of 21, while the second part involves the sum of two cube roots.

Simplifying the Cube Roots

To simplify the expression, we need to start by simplifying the cube roots. We can rewrite the cube roots as follows:

$\sqrt[3]{162} = \sqrt[3]{2 \cdot 81} = \sqrt[3]{2} \cdot \sqrt[3]{81} = \sqrt[3]{2} \cdot 9$

$\sqrt[3]{64} = \sqrt[3]{4 \cdot 16} = \sqrt[3]{4} \cdot \sqrt[3]{16} = \sqrt[3]{4} \cdot 2$

Simplifying the Sum of Cube Roots

Now that we have simplified the individual cube roots, we can simplify the sum of the cube roots. We can rewrite the sum as follows:

$\sqrt[3]{162} + \sqrt[3]{64} = \sqrt[3]{2} \cdot 9 + \sqrt[3]{4} \cdot 2$

Factoring Out Common Terms

To simplify the expression further, we can factor out common terms from the sum of the cube roots. We can rewrite the sum as follows:

$\sqrt[3]{162} + \sqrt[3]{64} = \sqrt[3]{2} \cdot 9 + \sqrt[3]{4} \cdot 2 = \sqrt[3]{2} \cdot (9 + \frac{2}{\sqrt[3]{2}})$

Simplifying the Expression

Now that we have simplified the sum of the cube roots, we can simplify the entire expression. We can rewrite the expression as follows:

$\sqrt{21} \left( \sqrt[3]{162} + \sqrt[3]{64} \right) = \sqrt{21} \cdot \sqrt[3]{2} \cdot (9 + \frac{2}{\sqrt[3]{2}})$

Applying the Distributive Property

To simplify the expression further, we can apply the distributive property. We can rewrite the expression as follows:

$\sqrt{21} \left( \sqrt[3]{162} + \sqrt[3]{64} \right) = \sqrt{21} \cdot \sqrt[3]{2} \cdot 9 + \sqrt{21} \cdot \sqrt[3]{2} \cdot \frac{2}{\sqrt[3]{2}}$

Simplifying the Expression

Now that we have applied the distributive property, we can simplify the expression further. We can rewrite the expression as follows:

$\sqrt{21} \left( \sqrt[3]{162} + \sqrt[3]{64} \right) = 9 \cdot \sqrt{21} \cdot \sqrt[3]{2} + 2 \cdot \sqrt{21}$

Factoring Out Common Terms

To simplify the expression further, we can factor out common terms. We can rewrite the expression as follows:

$\sqrt{21} \left( \sqrt[3]{162} + \sqrt[3]{64} \right) = \sqrt{21} \cdot (9 \cdot \sqrt[3]{2} + 2)$

Simplifying the Expression

Now that we have factored out common terms, we can simplify the expression further. We can rewrite the expression as follows:

$\sqrt{21} \left( \sqrt[3]{162} + \sqrt[3]{64} \right) = \sqrt{21} \cdot (9 \cdot \sqrt[3]{2} + 2)$

Conclusion

In this article, we have simplified the expression ${ \sqrt{21} \left( \sqrt[3]{162} + \sqrt[3]{64} \right) }$. We have applied various mathematical techniques and formulas to simplify the expression, including factoring out common terms, applying the distributive property, and simplifying radicals and exponents. The final simplified expression is $\sqrt{21} \cdot (9 \cdot \sqrt[3]{2} + 2)$.

Final Answer

The final answer is $\boxed{\sqrt{21} \cdot (9 \cdot \sqrt[3]{2} + 2)}$.

Step-by-Step Solution

Here is the step-by-step solution to the problem:

1. Simplify the cube roots: $\sqrt[3]{162} = \sqrt[3]{2} \cdot 9$ and $\sqrt[3]{64} = \sqrt[3]{4} \cdot 2$
2. Simplify the sum of the cube roots: $\sqrt[3]{162} + \sqrt[3]{64} = \sqrt[3]{2} \cdot 9 + \sqrt[3]{4} \cdot 2$
3. Factor out common terms: $\sqrt[3]{162} + \sqrt[3]{64} = \sqrt[3]{2} \cdot (9 + \frac{2}{\sqrt[3]{2}})$
4. Simplify the expression: $\sqrt{21} \left( \sqrt[3]{162} + \sqrt[3]{64} \right) = \sqrt{21} \cdot \sqrt[3]{2} \cdot (9 + \frac{2}{\sqrt[3]{2}})$
5. Apply the distributive property: $\sqrt{21} \left( \sqrt[3]{162} + \sqrt[3]{64} \right) = \sqrt{21} \cdot \sqrt[3]{2} \cdot 9 + \sqrt{21} \cdot \sqrt[3]{2} \cdot \frac{2}{\sqrt[3]{2}}$
6. Simplify the expression: $\sqrt{21} \left( \sqrt[3]{162} + \sqrt[3]{64} \right) = 9 \cdot \sqrt{21} \cdot \sqrt[3]{2} + 2 \cdot \sqrt{21}$
7. Factor out common terms: $\sqrt{21} \left( \sqrt[3]{162} + \sqrt[3]{64} \right) = \sqrt{21} \cdot (9 \cdot \sqrt[3]{2} + 2)$
8. Simplify the expression: $\sqrt{21} \left( \sqrt[3]{162} + \sqrt[3]{64} \right) = \sqrt{21} \cdot (9 \cdot \sqrt[3]{2} + 2)$

Final Answer

The final answer is $\boxed{\sqrt{21} \cdot (9 \cdot \sqrt[3]{2} + 2)}$.

Introduction

In our previous article, we simplified the expression ${ \sqrt{21} \left( \sqrt[3]{162} + \sqrt[3]{64} \right) }$. In this article, we will answer some frequently asked questions related to the simplification of this expression.

Q1: What is the first step in simplifying the expression?

A1: The first step in simplifying the expression is to simplify the cube roots. We can rewrite the cube roots as follows:

$\sqrt[3]{162} = \sqrt[3]{2 \cdot 81} = \sqrt[3]{2} \cdot \sqrt[3]{81} = \sqrt[3]{2} \cdot 9$

$\sqrt[3]{64} = \sqrt[3]{4 \cdot 16} = \sqrt[3]{4} \cdot \sqrt[3]{16} = \sqrt[3]{4} \cdot 2$

Q2: How do we simplify the sum of the cube roots?

A2: To simplify the sum of the cube roots, we can rewrite the sum as follows:

$\sqrt[3]{162} + \sqrt[3]{64} = \sqrt[3]{2} \cdot 9 + \sqrt[3]{4} \cdot 2$

Q3: What is the next step in simplifying the expression?

A3: The next step in simplifying the expression is to factor out common terms from the sum of the cube roots. We can rewrite the sum as follows:

$\sqrt[3]{162} + \sqrt[3]{64} = \sqrt[3]{2} \cdot (9 + \frac{2}{\sqrt[3]{2}})$

Q4: How do we apply the distributive property to simplify the expression?

A4: To apply the distributive property, we can rewrite the expression as follows:

$\sqrt{21} \left( \sqrt[3]{162} + \sqrt[3]{64} \right) = \sqrt{21} \cdot \sqrt[3]{2} \cdot (9 + \frac{2}{\sqrt[3]{2}})$

Q5: What is the final simplified expression?

A5: The final simplified expression is $\sqrt{21} \cdot (9 \cdot \sqrt[3]{2} + 2)$.

Q6: Can you provide a step-by-step solution to the problem?

A6: Yes, here is the step-by-step solution to the problem:

1. Simplify the cube roots: $\sqrt[3]{162} = \sqrt[3]{2} \cdot 9$ and $\sqrt[3]{64} = \sqrt[3]{4} \cdot 2$
2. Simplify the sum of the cube roots: $\sqrt[3]{162} + \sqrt[3]{64} = \sqrt[3]{2} \cdot 9 + \sqrt[3]{4} \cdot 2$
3. Factor out common terms: $\sqrt[3]{162} + \sqrt[3]{64} = \sqrt[3]{2} \cdot (9 + \frac{2}{\sqrt[3]{2}})$
4. Simplify the expression: $\sqrt{21} \left( \sqrt[3]{162} + \sqrt[3]{64} \right) = \sqrt{21} \cdot \sqrt[3]{2} \cdot (9 + \frac{2}{\sqrt[3]{2}})$
5. Apply the distributive property: $\sqrt{21} \left( \sqrt[3]{162} + \sqrt[3]{64} \right) = \sqrt{21} \cdot \sqrt[3]{2} \cdot 9 + \sqrt{21} \cdot \sqrt[3]{2} \cdot \frac{2}{\sqrt[3]{2}}$
6. Simplify the expression: $\sqrt{21} \left( \sqrt[3]{162} + \sqrt[3]{64} \right) = 9 \cdot \sqrt{21} \cdot \sqrt[3]{2} + 2 \cdot \sqrt{21}$
7. Factor out common terms: $\sqrt{21} \left( \sqrt[3]{162} + \sqrt[3]{64} \right) = \sqrt{21} \cdot (9 \cdot \sqrt[3]{2} + 2)$
8. Simplify the expression: $\sqrt{21} \left( \sqrt[3]{162} + \sqrt[3]{64} \right) = \sqrt{21} \cdot (9 \cdot \sqrt[3]{2} + 2)$

Conclusion

In this article, we have answered some frequently asked questions related to the simplification of the expression ${ \sqrt{21} \left( \sqrt[3]{162} + \sqrt[3]{64} \right) }$. We have provided step-by-step solutions to the problem and explained each step in detail.

Final Answer

The final answer is $\boxed{\sqrt{21} \cdot (9 \cdot \sqrt[3]{2} + 2)}$.