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The question in the image asks to write \( 81 \times 729 \) in index form.

1. First, express 81 and 729 as powers of 3:
   - \( 81 = 3^4 \) (since \( 81 = 3 \times 3 \times 3 \times 3 \)).
   - \( 729 = 3^6 \) (since \( 729 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \)).

2. Now, multiply these powers of 3:
   \[
   81 \times 729 = 3^4 \times 3^6
   \]

3. Using the rule of indices \( a^m \times a^n = a^{m+n} \), we can add the exponents:
   \[
   3^4 \times 3^6 = 3^{4+6} = 3^{10}
   \]

Therefore, the correct answer is **C: \( 3^{10} \)**.

Would you like further details or have any questions?

Here are 5 related questions:
1. How do you express a number as a power of a smaller base?
2. What are the rules of indices for multiplication?
3. How can you simplify \( a^m \times a^n \) in general?
4. What is the importance of base and exponent in index notation?
5. Can negative exponents appear when simplifying expressions?

**Tip**: When simplifying powers, always check for common bases before applying rules of indices.

Learn how to express 81 × 729 in index form using exponentiation and the law of indices. This problem explores simplifying powers of 3, suitable for Grades 8-10.

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