To convert the base-12 number $559_{12}$ and $819_{12}$ into base-10, we need to expand them using their place values based on base-12. Here's how:

1. Convert $559_{12}$:

The digits represent powers of 12, starting from the right:

$559_{12} = 5 \cdot 12^2 + 5 \cdot 12^1 + 9 \cdot 12^0$

Calculating each term:

$5 \cdot 12^2 = 5 \cdot 144 = 720$ $5 \cdot 12^1 = 5 \cdot 12 = 60$ $9 \cdot 12^0 = 9 \cdot 1 = 9$

Summing these:

$720 + 60 + 9 = 789$

So, $559_{12} = 789_{10}$.

2. Convert $819_{12}$:

$819_{12} = 8 \cdot 12^2 + 1 \cdot 12^1 + 9 \cdot 12^0$

Calculating each term:

$8 \cdot 12^2 = 8 \cdot 144 = 1152$ $1 \cdot 12^1 = 1 \cdot 12 = 12$ $9 \cdot 12^0 = 9 \cdot 1 = 9$

Summing these:

$1152 + 12 + 9 = 1173$

So, $819_{12} = 1173_{10}$.

Final Answer:

• $559_{12} = 789_{10}$
• $819_{12} = 1173_{10}$

Let me know if you want details or have questions! Here are 5 questions for practice:

1. Convert $ABC_{12}$ (where $A = 10$, $B = 11$, $C = 12$) to base-10.
2. How do you convert a base-10 number to base-12?
3. What is the base-10 equivalent of $34A_{12}$ where $A = 10$?
4. What is the highest digit possible in a base-12 system?
5. How does place value work in any positional numeral system?

Tip: When converting from any base to base-10, use powers of that base, starting from the rightmost digit.