Question

If `a^x * b^(3y) = root(4)(a^3 * b^-6)`, find the values of x and y, where a and b are different positive prime numbers.

Answer 1

Given:

`a^x * b^(3y) = root(4)(a^3 * b^-6)`

Where (a) and (b) are positive prime numbers and different.

Step-wise calculation:

1. Express the right-hand side with fractional exponents:

`root(4)(a^3 * b^-6) = (a^3 * b^-6)^(1/4)`

`root(4)(a^3 * b^-6) = a^(3/4) * b^(-6/4)`

`root(4)(a^3 * b^-6) = a^(3/4) * b^(-3/2)`

2. Since the bases (a) and (b) are prime and different, equate the powers of (a) and (b) on both sides:

For base (a):

`x = 3/4`

For base (b):

`3y = -3/2`

⇒ `y = -3/2 xx 1/3`

⇒ `y = -1/2`