What is the following product?

RootIndex 4 StartRoot 7 EndRoot times RootIndex 4 StartRoot 7 EndRoot times RootIndex 4 StartRoot 7 EndRoot times RootIndex 4 StartRoot 7 EndRoot
4 (RootIndex 4 StartRoot 4 EndRoot)
4 StartRoot 7 EndRoot
74
7

Let's break down the product and simplify the expression you provided. The key part is understanding that:

\[
\sqrt[4]{7} \times \sqrt[4]{7} \times \sqrt[4]{7} \times \sqrt[4]{7}
\]

is a multiplication of four fourth roots of 7.

### Step-by-step solution:

1. **Simplifying the product of the fourth roots:**
   \[
   \sqrt[4]{7} \times \sqrt[4]{7} \times \sqrt[4]{7} \times \sqrt[4]{7} = \left( \sqrt[4]{7} \right)^4 = 7
   \]

   Since multiplying four fourth roots of the same number results in the original number.

2. **Next part involves:** 
   \[
   4 \times \sqrt[4]{4}
   \]
   \(\sqrt[4]{4} = \sqrt{\sqrt{4}} = \sqrt{2}\), so:
   \[
   4 \times \sqrt[4]{4} = 4 \times \sqrt{2}
   \]

3. **Including the rest of the product:**
   You also have \(4 \sqrt{7}\), 74, and 7 remaining in the multiplication. Therefore, multiplying everything together:

   \[
   7 \times (4 \sqrt{2}) \times (4 \sqrt{7}) \times 74 \times 7
   \]

### Let's simplify the product step by step:

- First, multiply the constants: 
  \[
  7 \times 4 \times 4 \times 74 \times 7 = 7 \times 7 \times 4 \times 4 \times 74 = 49 \times 16 \times 74 = 784 \times 74 = 58016
  \]

- Then, handle the square roots:
  \[
  \sqrt{2} \times \sqrt{7} = \sqrt{14}
  \]

Thus, the final result is:
\[
58016 \sqrt{14}
\]

### Final Answer:
\[
58016 \sqrt{14}
\]

Would you like more details or have any questions?

Here are some related questions to practice:

1. What is the fourth root of 81?
2. How do you simplify the product of square roots?
3. If \(x = \sqrt[4]{7}\), how would you express \(x^5\)?
4. How do you multiply constants and square roots in an expression?
5. Can you simplify \(\sqrt[3]{8} \times \sqrt[3]{2}\)?

**Tip:** When multiplying roots with the same index, combine them under a single root to simplify.

What is the following product? RootIndex 4 StartRoot 7 EndRoot times RootIndex 4 StartRoot 7 EndRoot times RootIndex 4 StartRoot 7 EndRoot times RootIndex 4 StartRoot 7 EndRoot 4 (RootIndex 4 StartRoot 4 EndRoot) 4 StartRoot 7 EndRoot 74 7

Explore how to simplify the product of fourth roots and radicals in the expression. This step-by-step solution includes key concepts such as roots, exponents, and multiplication of radicals. The final result is expressed in a simplified form with square roots. Suitable for students in Grades 10-12.

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