Express 0.001 as a fraction: 1/1000. Write 1000 as a power of 10: 10^3. Therefore, 1/1000 = 10^-3. Setting 10^x = 10^-3 gives x = -3.

Rewrite 1/8 as a power of 2. We know 8 = 2^3, so 1/8 = 2^-3. The equation becomes 2^(x+2) = 2^-3. Equate the exponents: x + 2 = -3. Subtracting 2 gives x = -5.

Convert both bases to 3. Since 9 = 3^2 and 27 = 3^3, the equation is (3^2)^x = 3^3, or 3^(2x) = 3^3. Equating exponents gives 2x = 3, so x = 3/2 or 1.5.

Change both bases to 2. We know 4 = 2^2 and 8 = 2^3. Substitute these to get (2^2)^x = 2^3, which simplifies to 2^(2x) = 2^3. Equating exponents: 2x = 3, so x = 1.5.

Rewrite 125 with base 5. 125 is 5^3. The equation becomes 5^(2x) = 5^3. Equating the exponents gives 2x = 3. Dividing by 2 yields x = 1.5.

Write 27 as a power of 3. Since 3^3 = 27, substitute this into the equation: 3^(x-1) = 3^3. Equate the exponents: x - 1 = 3. Solving for x gives x = 4.

Express both sides with the same base. We know that 16 is 2 * 2 * 2 * 2, which is 2^4. So, 2^x = 2^4. Since the bases are identical, we can equate the exponents: x = 4.

Multiply top and bottom by the conjugate (sqrt(2) + 1). The denominator evaluates to (sqrt(2))^2 - 1^2 = 2 - 1 = 1. The result is simply the numerator, sqrt(2) + 1.

Multiply by the conjugate (sqrt(7) - sqrt(3)). The denominator becomes 7 - 3 = 4. The expression is 4(sqrt(7) - sqrt(3)) / 4. The 4s cancel, leaving sqrt(7) - sqrt(3).

First, combine into a single root: sqrt(2/8). This simplifies to sqrt(1/4). The square root of 1/4 is 1/2. Alternatively, sqrt(8) is 2 sqrt(2), so sqrt(2) / 2 sqrt(2) = 1/2.