Power or exponent denoted as a^x (a to the power x) means multiplying a with itself x times if x is an integer number. On the other hand, if x is a fractional number, for example 1/2, 1/3  it means square root of a, cube root of a.

2^3 = 2\times2\times2 = 8; 3 is an integer, so multiply 2 with 2 three times

4^{1/2} = (2\times2)^{1/2} = 2; 4 means multiply 2 with 2 two times, so its square root is just 2

27^{1/3} = (3\times3\times3)^{1/3} = 3; 27 means multiply 3 with 3 three times, so its cube root is just 3

16^{1/3} = (2\times2\times2\times2)^{1/3} = 2\sqrt[3] {2}; it means take one 2 out of the power for three 2, and calculate cube root of remaining 2

\sqrt[2] {2^6} = 8; is also a fractional power 2^{6/2} it means first compute 2⁶ which is 64, then take square root of 64 or (8×8)^1/2 which is 8. In other words, compute just 2^{6/2} = 2^3 = 8

Laws of Power:

a^m\cdot a^n = a^{m+n} \\ \\

\frac{a^m}{a^n} = a^{m-n} \\ \\

a^{-m} = \frac{1}{a^m} \\ \\

a^0 = 1 \\ \\

(a^m)^n = a^{mn} \\ \\

(ab)^m = a^mb^m \\ \\

1^m = 1 \\ \\

If a^x = a^y then x = y

Example:

Solve the equation \frac{2^x}{4^{x-1}} = 8 \\

\frac{2^x}{4^{x-1}} = 8 \\

or, \frac{2^x}{2^{2(x-1)}} = 8 \\

or, \frac{2^x}{2^{2x-2}} = 8 \\

or, 2^{x-2x+2} = 2^3 \\

or, x-2x+2 = 3 \\

or, x = -1

Example:

Solve the equation 3^{2x+1} - 2\cdot 3^{x+2} = 3^{x+2} \\

3^{2x+1} - 2\cdot 3^{x+2} = 3^{x+2} \\

or, 3^x(3^{x+1} - 2\cdot 3^2) = 3^{x+2} \\

or, 3^{x+1} - 18 = \frac{3^{x+2}}{3^x} \\

or, 3^{x+1} -18 = 3^{x+2-x} \\

or, 3^{x+1} = 18 + 3^2 = 27 \\

or, 3^{x+1} = 3^3 \\

or, x+1 = 3 \\

or, x = 2

Example:

Simplify the expression \frac{(ab)^2 x^3 y^{1/2}}{a^{-1}bx^2y} \\

\frac{(ab)^2 x^3 y^{1/2}}{a^{-1}bx^2y} \\

= a^{2+1}b^{2-1}x^{3-2}y^{1/2-1} \\

= a^3bxy^{-1/2}