You need to evaluate the solutions to the given equation, such that:

`sqrt(x^(lg (sqrt x))) = 10`

You need to remove the radical, hence, you should square both sides, such that:

`x^(lg (sqrt x)) = 100`

Taking decimal logarithms both sides, yields:

`lg (x^(lg (sqrt x))) = lg 100 `

Using the properties of logarithms yields:

`lg (sqrt x)*lg x = lg 10^2 => lg (sqrt x)*lg x = 2 lg 10`

Since `lg 10 = 1` yields:

`lg (sqrt x)*lg x = 2`

You should replace `(sqrt x)^2` for x, such that:

`lg (sqrt x)*lg (sqrt x)^2 = 2 => 2 lg (sqrt x)*lg (sqrt x) = 2`

Reducing duplicate factors yields:

`lg^2 (sqrt x) = 1 => lg (sqrt x) = +-1`

You need to solve the following equations, such that:

`{(lg (sqrt x) = 1),(lg (sqrt x) = -1):} => {((sqrt x) = 10),((sqrt x) = 1/10):}`

You need to solve the top equation, such that:

`sqrt x = 10`

Squaring both sides, yields:

`x_1 = 100`

You need to solve the bottom equation, such that:

`sqrt x = 1/10 => x_2 = 1/100`

You need to evaluate the product of the roots, such that:

`x_1*x_2 = 100*(1/100) `

Reducing duplicate factors yields:

`x_1*x_2 = 1`

**Hence, evaluating the product of the roots, under the given conditions, yields **`x_1*x_2 = 1.`