According to the exponent rule, any non-zero number raised to the power of zero equals 1.

By convention, 0^0 is often defined as 1 in combinatorics, although it can be considered indeterminate in other contexts.

The exponent zero indicates the multiplicative identity, meaning any non-zero number raised to the power of zero equals one.

Using the property of exponents, a^3 * a^(-2) = a^(3 - 2) = a^1, hence x = 1.

Substituting a = 2, we have 2^3 * b^2 = 64, which simplifies to 8b^2 = 64. Thus, b^2 = 8, leading to b = 4.

Taking the cube root of both sides, a = 27^(1/3) = 3.

Since 27 = 3^3, we have a = 3.

From a^3 = b^2, we can express b in terms of a as b = a^(3/2).

According to the laws of exponents, when multiplying like bases, we add the exponents: p = m + n.

The property of exponents states that when multiplying like bases, you add the exponents: m + n = p.

If a = b, then a^x = b^y implies x must equal y for the equality to hold.

Substituting a = b^k into a^x = b^y gives (b^k)^x = b^y, leading to kx = y, hence x = ky.

From a^x = b^y, taking log_a on both sides gives log_a(b^y) = x, hence y * log_a(b) = x, thus log_a(b) = y/x.

The rank of matrix B is the number of non-zero rows in its row echelon form, which is 3.

The determinant of B is calculated as (2*7) - (3*5) = 14 - 15 = -1.

The power set of a set with n elements has 2^n subsets. Here, n = 4, so the power set has 2^4 = 16 subsets.

The number of subsets of a set with n elements is 2^n. Here, n=2, so the number of subsets is 2^2 = 4.

The power set of B = {a, b} is {∅, {a}, {b}, {a, b}}.

det(B) = (2*4) - (3*1) = 8 - 3 = 5.

The number of selections is given by C(4,2) = 6.

B is the brother of D as they share the same parent, C.

C · D = 0*3 + 1*4 + 2*5 = 0 + 4 + 10 = 14.

C · D = 3*0 + (-1)*4 + 2*(-1) = 0 - 4 - 2 = -6.

C · D = 3*0 + (-1)*4 + 2*(-3) = 0 - 4 - 6 = -10.

C · D = 3*4 + (-1)*2 + 2*1 = 12 - 2 + 2 = 12.

C · D = 3*0 + (-2)*4 + 1*(-3) = 0 - 8 - 3 = -11.

Determinant of C = (0*0) - (1*1) = 0 - 1 = -1.

The determinant of C is 0 because it has a row of zeros.

Using the determinant formula for 3x3 matrices, det(C) = 1(3*0 - 1*1) - 0 + 2(-1*1 - 3*2) = 0 - 0 - 12 = -12.

The determinant of an upper triangular matrix is the product of its diagonal elements. Here, det(C) = 1 * 1 * 1 = 1.