find the value of the base so the equation is correct: 6651+115=10066

To find the value of the base \( b \) so that the equation \( 6651_b + 115_b = 10066_b \) holds true, we'll follow these steps:

1. Convert each number from base \( b \) to base 10.
2. Solve for \( b \) such that the sum of the first two numbers equals the third number in base 10.

### Step-by-Step Solution:

1. **Convert \( 6651_b \) to base 10:**
   \[
   6651_b = 6b^3 + 6b^2 + 5b^1 + 1b^0
   \]
   
2. **Convert \( 115_b \) to base 10:**
   \[
   115_b = 1b^2 + 1b^1 + 5b^0
   \]

3. **Convert \( 10066_b \) to base 10:**
   \[
   10066_b = 1b^5 + 0b^4 + 0b^3 + 6b^2 + 6b^1 + 6b^0
   \]

### Form the Equation:

Since \( 6651_b + 115_b = 10066_b \) must hold true, we get:
\[
(6b^3 + 6b^2 + 5b + 1) + (1b^2 + 1b + 5) = 1b^5 + 0b^4 + 0b^3 + 6b^2 + 6b + 6
\]

Combine like terms on the left-hand side:
\[
6b^3 + 7b^2 + 6b + 6 = b^5 + 6b^2 + 6b + 6
\]

Now simplify the equation by subtracting \( (6b^2 + 6b + 6) \) from both sides:
\[
6b^3 + 7b^2 + 6b + 6 - 6b^2 - 6b - 6 = b^5
\]
\[
6b^3 + b^2 = b^5
\]

Rearrange the equation to solve for \( b \):
\[
b^5 - 6b^3 - b^2 = 0
\]

Factor out the common term \( b^2 \):
\[
b^2 (b^3 - 6b - 1) = 0
\]

Since \( b \neq 0 \) in this context (because base cannot be zero), we focus on:
\[
b^3 - 6b - 1 = 0
\]

### Solving the Cubic Equation:

To find the value of \( b \), we solve the cubic equation \( b^3 - 6b - 1 = 0 \). This can be done using numerical methods or trial and error.

By trial, we check possible values:
- For \( b = 2 \): \( 2^3 - 6(2) - 1 = 8 - 12 - 1 = -5 \) (too low)
- For \( b = 3 \): \( 3^3 - 6(3) - 1 = 27 - 18 - 1 = 8 \) (too high)
- For \( b = 2.5 \): \( 2.5^3 - 6(2.5) - 1 = 15.625 - 15 - 1 = -0.375 \) (closer)
- For \( b = 2.7 \): \( 2.7^3 - 6(2.7) - 1 = 19.683 - 16.2 - 1 = 2.483 \) (still too high)

Refining the value within the interval, we find that the solution is approximately \( b \approx 2.45 \), but typically base is an integer. Let's check for more precise integer values.

### Check if approximate or integer:

Using exact factorization or numerical methods, \( b \approx 2.45 \).

Therefore, solving for the base is more straightforward if \( b \) can be approximated as 2.45. In a strict integer base context, no exact integer solution satisfies this equation closely without more computational methods.

Would you like more detailed steps or have other questions?

### Suggested Questions:
1. How do we convert numbers from one base to another?
2. Can you explain solving cubic equations in more detail?
3. What are some methods to solve polynomial equations?
4. How does the base affect number representation?
5. What are common applications of different bases in math and computing?

### Tip:
Understanding how to convert between bases can greatly help in grasping various number systems, especially in computer science where binary, octal, and hexadecimal systems are prevalent.

Discover how to find the base value \( b \) for the equation 6651_b + 115_b = 10066_b. Learn step-by-step methods for converting numbers between bases and solving cubic equations. Suitable for advanced high school students.

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