To decompose the number 735 in base 8 using exponents, we follow the steps below.

Step 1: Understanding the representation

The number 735 in base 8 can be written as a sum of powers of 8, since it represents:

$735_8 = 7 \times 8^2 + 3 \times 8^1 + 5 \times 8^0$

Step 2: Break it down

We decompose each digit using exponents:

• The first digit, 7, is in the $8^2$ place.
• The second digit, 3, is in the $8^1$ place.
• The third digit, 5, is in the $8^0$ place.

So, this gives us the following expression:

$735_8 = 7 \times 8^2 + 3 \times 8^1 + 5 \times 8^0$

Step 3: Simplify each term

• $7 \times 8^2 = 7 \times 64 = 448$
• $3 \times 8^1 = 3 \times 8 = 24$
• $5 \times 8^0 = 5 \times 1 = 5$

Step 4: Final sum

Adding these together:

$735_8 = 448 + 24 + 5 = 477_{10}$

Thus, the base-8 number $735_8$ decomposes to $7 \times 8^2 + 3 \times 8^1 + 5 \times 8^0$, which equals $477_{10}$ in base 10.

Would you like more details on base conversions, or do you have any questions?

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Here are 5 related questions:

1. How do you convert a base-10 number back to base 8?
2. What is the process to convert other bases, like base 16 or base 2, into base 10?
3. How does base 8 compare to binary (base 2) in terms of practical applications?
4. What are the steps to convert a decimal fraction into base 8?
5. How can you quickly recognize whether a number is in base 8 from its