Let's break down and solve the expression from the image:

$5^3 + 4^3 \cdot 4^7 - \frac{12^{10}}{3^{10}}$

Step-by-step solution:

1. Simplify each part:

   • $5^3 = 5 \times 5 \times 5 = 125$
   • $4^3 = 4 \times 4 \times 4 = 64$
   • $4^7 = 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 = 16384$
   • The fraction $\frac{12^{10}}{3^{10}}$ can be rewritten as: $\frac{12^{10}}{3^{10}} = \left(\frac{12}{3}\right)^{10} = 4^{10}$ Now, calculate $4^{10} = 1048576$.

2. Now, substitute the simplified terms into the expression: $5^3 + 4^3 \cdot 4^7 - \frac{12^{10}}{3^{10}} = 125 + 64 \cdot 16384 - 1048576$

3. Perform multiplication and subtraction:

   • First, calculate $64 \cdot 16384 = 1048576$
   • So, the expression becomes: $125 + 1048576 - 1048576$
   • This simplifies to: $125 + 0 = 125$

Final Answer:

$\boxed{125}$

Would you like more details or have any questions?

Here are 5 related questions to explore:

1. How would this expression change if you used different base values for the powers?
2. What if the fraction was reversed, i.e., $\frac{3^{10}}{12^{10}}$?
3. Can you simplify $4^{10}$ without using a calculator, through factorization?
4. How would the solution differ if the exponent on $12$ and $3$ was not equal, say $12^8$ over $3^{10}$?
5. How could you verify the result using logarithmic properties?

Tip: When simplifying powers and exponents in fractions, try to find common bases to simplify the expression more easily!