What Is The Following Product? 5 3 ⋅ 2 \sqrt[3]{5} \cdot \sqrt{2} 3 5 ⋅ 2 A. 10 6 \sqrt[6]{10} 6 10 B. 200 6 \sqrt[6]{200} 6 200 C. 500 6 \sqrt[6]{500} 6 500 D. 100000 6 \sqrt[6]{100000} 6 100000

Mar 13, 2025 by ADMIN 208 views

$What is the Following Product? $\sqrt[3]{5} \cdot \sqrt{2}$$

Understanding the Problem

The given problem involves the multiplication of two radical expressions: $\sqrt[3]{5}$ and $\sqrt{2}$ . To solve this, we need to understand the properties of radicals and how to multiply them. The expression $\sqrt[3]{5}$ represents the cube root of 5, while $\sqrt{2}$ represents the square root of 2.

Properties of Radicals

Before we proceed with the multiplication, let's review some key properties of radicals:

The product of two radical expressions with the same index can be simplified by multiplying the radicands (the numbers inside the radical sign).
The product of two radical expressions with different indices can be simplified by finding the least common multiple (LCM) of the indices and then simplifying the resulting expression.

Multiplying Radical Expressions

Now, let's apply these properties to the given problem. We need to multiply $\sqrt[3]{5}$ and $\sqrt{2}$ :

$\sqrt[3]{5} \cdot \sqrt{2} = \sqrt[3]{5 \cdot 2}$

Simplifying the Expression

The next step is to simplify the expression inside the radical sign. We can do this by multiplying 5 and 2:

$5 \cdot 2 = 10$

So, the expression becomes:

$\sqrt[3]{10}$

Evaluating the Expression

Now that we have simplified the expression, we need to evaluate it. The cube root of 10 is a number that, when multiplied by itself three times, gives 10. Let's find this number:

$\sqrt[3]{10} = x$

$x^3 = 10$

$x = \sqrt[3]{10}$

Comparing with the Options

Now, let's compare the result with the options provided:

A. $\sqrt[6]{10}$ B. $\sqrt[6]{200}$ C. $\sqrt[6]{500}$ D. $\sqrt[6]{100000}$

Conclusion

Based on our calculation, the correct answer is:

$\boxed{\sqrt[6]{100000}}$

This is because $\sqrt[6]{100000} = \sqrt[6]{(10)^3} = \sqrt[6]{10^3} = \sqrt[6]{10}^3 = \sqrt[3]{10}$ .

Final Answer

The final answer is $\boxed{\sqrt[6]{100000}}$ .

Introduction

Radical expressions are a fundamental concept in mathematics, and understanding them is crucial for solving various mathematical problems. In this article, we will provide a comprehensive Q&A guide to help you understand radical expressions and how to work with them.

Q1: What is a Radical Expression?

A radical expression is a mathematical expression that contains a root or a power of a number. It is denoted by a symbol called a radical sign, which is usually represented by a horizontal line above the root or power.

Q2: What is the Difference between a Square Root and a Cube Root?

A square root is a radical expression that represents the number that, when multiplied by itself, gives a specified value. For example, $\sqrt{16} = 4$ because $4 \times 4 = 16$ . A cube root, on the other hand, is a radical expression that represents the number that, when multiplied by itself three times, gives a specified value. For example, $\sqrt[3]{27} = 3$ because $3 \times 3 \times 3 = 27$ .

Q3: How Do I Simplify a Radical Expression?

To simplify a radical expression, you need to find the largest perfect square or perfect cube that divides the radicand (the number inside the radical sign). You can then take the square root or cube root of this perfect square or perfect cube and simplify the expression.

Q4: What is the Product of Two Radical Expressions?

The product of two radical expressions is a new radical expression that is the result of multiplying the two original expressions. To multiply two radical expressions, you need to multiply the radicands (the numbers inside the radical signs) and then simplify the resulting expression.

Q5: How Do I Multiply Radical Expressions with Different Indices?

When multiplying radical expressions with different indices, you need to find the least common multiple (LCM) of the indices and then simplify the resulting expression.

Q6: What is the Quotient of Two Radical Expressions?

The quotient of two radical expressions is a new radical expression that is the result of dividing the two original expressions. To divide two radical expressions, you need to divide the radicands (the numbers inside the radical signs) and then simplify the resulting expression.

Q7: How Do I Simplify a Radical Expression with a Fractional Index?

A radical expression with a fractional index is a radical expression that has a root or power that is a fraction. To simplify such an expression, you need to find the square root or cube root of the radicand and then simplify the resulting expression.

Q8: What is the Difference between a Rational and Irrational Number?

A rational number is a number that can be expressed as a ratio of two integers, while an irrational number is a number that cannot be expressed as a ratio of two integers. Radical expressions can result in both rational and irrational numbers.

Q9: How Do I Use Radical Expressions in Real-World Applications?

Radical expressions have numerous real-world applications, including physics, engineering, and finance. They are used to model and solve problems involving growth and decay, motion, and optimization.

Q10: What are Some Common Mistakes to Avoid when Working with Radical Expressions?

Some common mistakes to avoid when working with radical expressions include:

Not simplifying the radicand before taking the root or power
Not finding the least common multiple (LCM) of the indices when multiplying radical expressions with different indices
Not simplifying the resulting expression after multiplying or dividing radical expressions

Conclusion

Radical expressions are a fundamental concept in mathematics, and understanding them is crucial for solving various mathematical problems. By following the Q&A guide provided in this article, you will be able to understand radical expressions and how to work with them. Remember to simplify the radicand before taking the root or power, find the least common multiple (LCM) of the indices when multiplying radical expressions with different indices, and simplify the resulting expression after multiplying or dividing radical expressions.