Answer:
2x²√13
Step-by-step explanation:
Step 1:  When there is multiplication under a square root sign (also called a radical sign), you can rewrite the expression as the product of two square roots, like this...
√(52x^4)   can be rewritten as     (√52)(√x^4)
Step 2:  x^4 = x²x², so it's the same thing as saying (x²)², or "x-squared, squared".  Taking the square root of it leaves just x-squared, so this simplifies to 
    (√52)x²
Step 3:  When taking the square root of a number that is not a perfect square, you need to do a factor tree to see if you can break the number up into a product of perfect squares, or a perfect square multiplied by a prime number.
52 is an even number, so we an at least divide by 2.   
         52/2  is 26, so we have
                   (26)(2)
 2 is a prime number, so we don't break that down any further.
26 is even so it is also divisible by 2, 
    26/2 is 13, so we have
      (13)(2)(2)
   which we can write as (13)(4)     (groups pairs of numbers together)
 
(2)(2) is 4, which results in a perfect square, so we can rewrite √52 as
       (√4)(√13),       since √4  = 2, we simplify this expression to 
         2√13
So √(52x^4) breaks down to (√52)(√x^4), which further breaks down into
   (2√13)(x²)
We simplify the expression to 2x²√13