We can simplify this equation by using the properties of exponents and radicals.
First, we can simplify the square roots:
√25x^n × √20 = √(25x^n * 20)
Next, we can simplify the right side of the equation:
10x⁵√5x = 10 * x^5 * √(5x)
Now we can set the two sides of the equation equal to each other:
√(25x^n * 20) = 10 * x^5 * √(5x)
We can simplify the square root on the left side by factoring out 5:
√(25 * 5 * x^n * 4) = 10 * x^5 * √(5x)
Simplifying further:
5 * √(5 * x^n * 4) = 10 * x^5 * √(5x)
Now we can simplify the square root on the left side:
5 * √(20x^n) = 10 * x^5 * √(5x)
We can simplify the coefficient on the left side:
√(20x^n) = 2 * x^5 * √(5x)
Now we can square both sides of the equation to eliminate the square root:
20x^n = 4x^10 * 5x
Simplifying:
20x^n = 20x^11
Dividing both sides by 20:
x^n = x^11
Now we can solve for n by using the property that x^a / x^b = x^(a-b):
n = 11 - 1
n = 10
Therefore, the value of n is 10.