To simplify these expressions, we can use the properties of exponents.

For the first expression, 125^6:5^3, we can rewrite 125 as 5^3. So, the expression becomes (5^3)^6:5^3. Using the power of a power property, we can simplify this to 5^(3*6):5^3. This further simplifies to 5^18:5^3.

Using the quotient of powers property, we subtract the exponents: 5^(18-3) = 5^15.

So, 125^6:5^3 simplifies to 5^15.

For the second expression, 27^15:9^3, we can rewrite 27 as 3^3. So, the expression becomes (3^3)^15:9^3. Using the power of a power property, we can simplify this to 3^(3*15):9^3. This further simplifies to 3^45:9^3.

Using the quotient of powers property, we subtract the exponents: 3^(45-3) = 3^42.

So, 27^15:9^3 simplifies to 3^42.

For the third expression, 16^6:2^6, we can rewrite 16 as 2^4. So, the expression becomes (2^4)^6:2^6. Using the power of a power property, we can simplify this to 2^(4*6):2^6. This further simplifies to 2^24:2^6.

Using the quotient of powers property, we subtract the exponents: 2^(24-6) = 2^18.

So, 16^6:2^6 simplifies to 2^18.

For the fourth expression, 81^10:27^2, we can rewrite 81 as 3^4. So, the expression becomes (3^4)^10:27^2. Using the power of a power property, we can simplify this to 3^(4*10):27^2. This further simplifies to 3^40:27^2.

Using the quotient of powers property, we subtract the exponents: 3^(40-2) = 3^38.

So, 81^10:27^2 simplifies to 3^38.