Question

How do you solve `(25x ^ { - \frac { 1} { 2} } y ^ { \frac { 2} { 2} } ) ^ { \frac { 1} { 2} }`?

Answer 1

See a solution process below:

Explanation:

First, use these rules for exponents to eliminate the outer exponent:

`a = a^color(red)(1)` and `(x^color(red)(a))^color(blue)(b) = x^(color(red)(a) xx color(blue)(b))`

`(25x^(-1/2)y^(2/2))^(1/2) => (25^color(red)(1)x^color(red)(-1/2)y^color(red)(2/2))^color(blue)(1/2) => 25^(color(red)(1)xxcolor(blue)(1/2))x^(color(red)(-1/2)xxcolor(blue)(1/2))y^(color(red)(2/2)xxcolor(blue)(1/2)) =>`

`25^(1/2)x^-1/4y^(2/4) => 25^(1/2)x^(-1/4)y^(1/2)`

We can now use this rule for exponents/radicals to evaluate the constant:

`x^(1/color(red)(n)) = root(color(red)(n))(x)`

`25^color(red)(1/2)x^(-1/4)y^(1/2) => sqrt(25)x^(-1/4)y^(1/2) => 5x^(-1/4)y^(1/2)`

Next, we can use this rule for exponents to eliminate the negative exponent:

`x^color(red)(a) = 1/x^color(red)(-a)`

`5x^color(red)(-1/4)y^(1/2) => (5y^(1/2))/x^-color(red)(-1/4) => (5y^(1/2))/x^(1/4)`