An exponent of 1/4 indicates the fourth root of the base. The fourth root of 16 is 2, since 2 * 2 * 2 * 2 equals 16.

We need to find a number whose cube is 216. Since 6 * 6 * 6 equals exactly 216, the cube root of 216 is 6.

Resolving the decimals first outputs 0.000001 (which amounts to exactly 10^-6). Multiplying 10^-6 and 10^7 combines the given exponents safely to construct a finalized value of 10.

Using the quotient rule, we keep the base of 3 and subtract the denominator's exponent from the numerator's exponent. The calculation 7 - 2 leaves us with 3^5.

Using the quotient property, merge the terms into one square root: sqrt(50 / 2). This simplifies to sqrt(25), which has a value of 5.

According to the zero exponent rule, any non-zero real number raised to the power of zero is always equal to 1. Therefore, 5^0 equals 1.

Evaluate the numerical and variable parts separately. The square root of 16 is 4, and the square root of x^4 is x^2. This gives the exact simplified form 4x^2.

Calculating the individual values produces: (0.09)^2 = 0.0081, (1-0.9)^2 = (0.1)^2 = 0.01, and 1 - (0.9)^2 = 1 - 0.81 = 0.19. The number 0.0081 operates as the minimal absolute magnitude making it closest to zero.

Any non-zero number raised to the power of zero is always equal to 1. The negative sign is inside the parentheses, meaning the entire base of -5 is raised to the zero power, resulting in 1.

To rewrite a term with a negative exponent, place it in the denominator with a positive exponent. Therefore, x^-2 is equivalent to 1/x^2.