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

Understanding the Problem

The given problem involves the multiplication of four fourth roots of 7. To solve this, we need to understand the properties of exponents and roots. The fourth root of a number is equivalent to raising that number to the power of 1/4. In this case, we have $\sqrt[4]{7}$, which is equivalent to $7^{1/4}$.

Breaking Down the Problem

Let's break down the given expression:

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

We can rewrite each fourth root as $7^{1/4}$:

$7^{1/4} \cdot 7^{1/4} \cdot 7^{1/4} \cdot 7^{1/4}$

Using Exponent Rules

When multiplying numbers with the same base, we can add their exponents. In this case, we have the same base (7) and we are multiplying four numbers with the same exponent (1/4). Therefore, we can add the exponents:

$7^{1/4} \cdot 7^{1/4} \cdot 7^{1/4} \cdot 7^{1/4} = 7^{1/4 + 1/4 + 1/4 + 1/4}$

Simplifying the Exponent

To add the exponents, we need to find a common denominator. In this case, the common denominator is 4. Therefore, we can rewrite each exponent as a fraction with a denominator of 4:

$7^{1/4 + 1/4 + 1/4 + 1/4} = 7^{4/4 + 4/4 + 4/4 + 4/4}$

Now, we can add the numerators:

$7^{4/4 + 4/4 + 4/4 + 4/4} = 7^{16/4}$

Simplifying the Fraction

We can simplify the fraction by dividing the numerator by the denominator:

$7^{16/4} = 7^{4}$

Conclusion

Therefore, the product of four fourth roots of 7 is equivalent to $7^4$. This is option C.

Answer

The correct answer is C. $7^4$.

Explanation

The other options are incorrect because:

• Option A is incorrect because it is missing the exponent 4.
• Option B is incorrect because it is raising 7 to the power of 2, not 4.
• Option D is incorrect because it is simply 7, not $7^4$.

Final Answer

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Q: What is the product of four fourth roots of 7?

A: The product of four fourth roots of 7 is equivalent to $7^4$.

Q: Why can we add the exponents when multiplying numbers with the same base?

A: When multiplying numbers with the same base, we can add their exponents because of the properties of exponents. Specifically, when we multiply numbers with the same base, we can add their exponents.

Q: How do we add exponents with different denominators?

A: To add exponents with different denominators, we need to find a common denominator. In this case, the common denominator is 4. Therefore, we can rewrite each exponent as a fraction with a denominator of 4.

Q: What is the common denominator for adding exponents?

A: The common denominator for adding exponents is the least common multiple (LCM) of the denominators. In this case, the LCM of 4 is 4.

Q: How do we simplify a fraction with a large numerator and denominator?

A: To simplify a fraction with a large numerator and denominator, we can divide the numerator by the denominator. In this case, we can simplify the fraction $16/4$ by dividing the numerator by the denominator.

Q: What is the simplified fraction?

A: The simplified fraction is 4.

Q: What is the final answer to the problem?

A: The final answer is C. $7^4$.

Q: Why is option A incorrect?

A: Option A is incorrect because it is missing the exponent 4.

Q: Why is option B incorrect?

A: Option B is incorrect because it is raising 7 to the power of 2, not 4.

Q: Why is option D incorrect?

A: Option D is incorrect because it is simply 7, not $7^4$.

Q: What is the key concept in solving this problem?

A: The key concept in solving this problem is understanding the properties of exponents and how to add exponents with different denominators.

Q: How can I apply this concept to other problems?

A: You can apply this concept to other problems by recognizing when you can add exponents and simplifying fractions with large numerators and denominators.

Q: What are some common mistakes to avoid when solving problems like this?

A: Some common mistakes to avoid when solving problems like this include:

• Not recognizing when you can add exponents
• Not simplifying fractions with large numerators and denominators
• Not checking your work for errors

Q: How can I practice solving problems like this?

A: You can practice solving problems like this by working through examples and exercises in your textbook or online resources. You can also try solving problems on your own and checking your work with a calculator or online tool.