What Is The Following Product?$\sqrt[4]{7} \cdot \sqrt[4]{7} \cdot \sqrt[4]{7} \cdot \sqrt[4]{7}$A. $4(\sqrt[4]{7}$\]B. $4 \sqrt{7}$C. $7^4$D. 7

When dealing with mathematical expressions involving radicals, it's essential to understand the properties and rules that govern their behavior. In this article, we will explore the product of radicals and how it relates to the given expression $\sqrt[4]{7} \cdot \sqrt[4]{7} \cdot \sqrt[4]{7} \cdot \sqrt[4]{7}$.

What is a Radical?

A radical is a mathematical expression that represents a number that can be expressed as the product of a whole number and a square root. The radical symbol, denoted by $\sqrt[n]{x}$, represents the nth root of x. For example, $\sqrt[4]{7}$ represents the fourth root of 7.

Properties of Radicals

Radicals have several properties that are essential to understand when working with them. One of the most important properties is the product rule, which states that the product of two or more radicals is equal to the radical of the product of the numbers inside the radicals.

The Product Rule

The product rule can be expressed as:

$$\sqrt[n]{x} \cdot \sqrt[n]{y} = \sqrt[n]{xy}$$

This means that when we multiply two or more radicals with the same index, we can combine them into a single radical with the product of the numbers inside.

Applying the Product Rule

Now that we have a clear understanding of the product rule, let's apply it to the given expression $\sqrt[4]{7} \cdot \sqrt[4]{7} \cdot \sqrt[4]{7} \cdot \sqrt[4]{7}$.

Using the product rule, we can combine the four radicals into a single radical with the product of the numbers inside:

$$\sqrt[4]{7} \cdot \sqrt[4]{7} \cdot \sqrt[4]{7} \cdot \sqrt[4]{7} = \sqrt[4]{7 \cdot 7 \cdot 7 \cdot 7}$$

Simplifying the Expression

Now that we have combined the radicals, we can simplify the expression by evaluating the product inside the radical:

$$\sqrt[4]{7 \cdot 7 \cdot 7 \cdot 7} = \sqrt[4]{7^4}$$

Evaluating the Radical

The final step is to evaluate the radical by finding the fourth root of $7^4$. Since the index of the radical is 4, we can simply raise 7 to the power of 4 to find the value inside the radical:

$$\sqrt[4]{7^4} = 7^{\frac{4}{4}} = 7^1 = 7$$

Conclusion

In conclusion, the product of radicals can be simplified using the product rule, which states that the product of two or more radicals is equal to the radical of the product of the numbers inside the radicals. By applying this rule to the given expression $\sqrt[4]{7} \cdot \sqrt[4]{7} \cdot \sqrt[4]{7} \cdot \sqrt[4]{7}$, we can simplify it to $7$. This demonstrates the importance of understanding the properties and rules of radicals in mathematics.

Answer

The correct answer is D. 7.

Discussion

This problem requires a deep understanding of the properties and rules of radicals, including the product rule. It also requires the ability to apply mathematical concepts to simplify complex expressions. The correct answer, D. 7, is the result of applying the product rule and simplifying the expression using the properties of radicals.

Related Topics

• Radicals and their properties
• Product rule for radicals
• Simplifying expressions involving radicals
• Mathematical concepts and their applications

Further Reading

For more information on radicals and their properties, including the product rule, we recommend the following resources:

• Khan Academy: Radicals and Rational Exponents
• Mathway: Radicals and Exponents
• Wolfram MathWorld: Radicals and Roots

In our previous article, we explored the product of radicals and how it relates to the given expression $\sqrt[4]{7} \cdot \sqrt[4]{7} \cdot \sqrt[4]{7} \cdot \sqrt[4]{7}$. We also discussed the properties and rules of radicals, including the product rule. In this article, we will answer some frequently asked questions about radicals and their properties.

Q: What is a radical?

A: A radical is a mathematical expression that represents a number that can be expressed as the product of a whole number and a square root. The radical symbol, denoted by $\sqrt[n]{x}$, represents the nth root of x.

Q: What is the product rule for radicals?

A: The product rule for radicals states that the product of two or more radicals is equal to the radical of the product of the numbers inside the radicals. This can be expressed as:

$$\sqrt[n]{x} \cdot \sqrt[n]{y} = \sqrt[n]{xy}$$

Q: How do I simplify an expression involving radicals?

A: To simplify an expression involving radicals, you can use the product rule and the properties of radicals. For example, if you have the expression $\sqrt[4]{7} \cdot \sqrt[4]{7} \cdot \sqrt[4]{7} \cdot \sqrt[4]{7}$, you can combine the radicals using the product rule and simplify the expression to $7$.

Q: What is the difference between a radical and an exponent?

A: A radical and an exponent are both used to represent powers of a number, but they are used in different ways. A radical represents the nth root of a number, while an exponent represents the power to which a number is raised. For example, $\sqrt[4]{7}$ represents the fourth root of 7, while $7^4$ represents 7 raised to the power of 4.

Q: Can I use the product rule for radicals with different indices?

A: No, the product rule for radicals only applies to radicals with the same index. If you have radicals with different indices, you cannot use the product rule to combine them.

Q: How do I evaluate a radical expression?

A: To evaluate a radical expression, you need to find the value inside the radical. For example, if you have the expression $\sqrt[4]{7^4}$, you can evaluate it by finding the fourth root of $7^4$, which is equal to 7.

Q: What are some common mistakes to avoid when working with radicals?

A: Some common mistakes to avoid when working with radicals include:

• Not using the product rule for radicals when combining radicals with the same index
• Using the product rule for radicals with different indices
• Not evaluating the radical expression correctly
• Not simplifying the expression correctly

Conclusion

In conclusion, radicals and their properties are an essential part of mathematics. Understanding the product rule for radicals and how to simplify expressions involving radicals can help you solve mathematical problems with confidence. By avoiding common mistakes and using the product rule correctly, you can master the art of working with radicals.

Related Topics

• Radicals and their properties
• Product rule for radicals
• Simplifying expressions involving radicals
• Mathematical concepts and their applications

Further Reading

For more information on radicals and their properties, including the product rule, we recommend the following resources:

• Khan Academy: Radicals and Rational Exponents
• Mathway: Radicals and Exponents
• Wolfram MathWorld: Radicals and Roots

By understanding the properties and rules of radicals, including the product rule, you can simplify complex expressions and solve mathematical problems with confidence.