What is $\frac{1357_{9}}{100_{4}}-2460_{8}+5678_{9}$? Express your answer in base 10.

Hello Guest! Let's solve the problem step by step. The problem asks us to evaluate the expression:

$\frac{1357_{9}}{100_{4}} - 2460_{8} + 5678_{9}$

and express the answer in base 10. Here’s the plan:

Step-by-Step Plan

1. Convert all numbers to base 10:

   • Convert $1357_9$ to base 10.
   • Convert $100_4$ to base 10.
   • Convert $2460_8$ to base 10.
   • Convert $5678_9$ to base 10.

2. Evaluate the division:

   • Divide the base 10 value of $1357_9$ by the base 10 value of $100_4$.

3. Perform the subtraction and addition:

   • Subtract the base 10 value of $2460_8$ from the result of the division.
   • Add the base 10 value of $5678_9$ to the result.

4. Present the final answer in base 10.

Let's begin by converting each number to base 10.

1. Convert $1357_9$ to base 10

The formula for converting from base $b$ to base 10 is:

$d_n b^n + d_{n-1} b^{n-1} + \ldots + d_0 b^0$

For $1357_9$:

$1 \times 9^3 + 3 \times 9^2 + 5 \times 9^1 + 7 \times 9^0$

2. Convert $100_4$ to base 10

$1 \times 4^2 + 0 \times 4^1 + 0 \times 4^0$

3. Convert $2460_8$ to base 10

$2 \times 8^3 + 4 \times 8^2 + 6 \times 8^1 + 0 \times 8^0$

4. Convert $5678_9$ to base 10

$5 \times 9^3 + 6 \times 9^2 + 7 \times 9^1 + 8 \times 9^0$

Now, let's calculate each value. Guest, here is the detailed solution:

1. Convert each number to base 10:

   • $1357_9 = 1 \times 9^3 + 3 \times 9^2 + 5 \times 9^1 + 7 \times 9^0 = 1024$
   • $100_4 = 1 \times 4^2 + 0 \times 4^1 + 0 \times 4^0 = 16$
   • $2460_8 = 2 \times 8^3 + 4 \times 8^2 + 6 \times 8^1 + 0 \times 8^0 = 1328$
   • $5678_9 = 5 \times 9^3 + 6 \times 9^2 + 7 \times 9^1 + 8 \times 9^0 = 4202$

2. Divide the first two results:

   • $\frac{1024}{16} = 64$

3. Subtract and add as required:

   • $64 - 1328 + 4202 = 2938$

Final Answer:

This is the value of the original expression in base 10.