To find the value of x in the expression (x + xlog10x)^5, given that the third term of the expansion is 106, we can use the binomial theorem. The binomial theorem states that for any positive integer n:

(x + y)^n = C(n,0) * x^n * y^0 + C(n,1) * x^(n-1) * y^1 + C(n,2) * x^(n-2) * y^2 + ... + C(n,n-1) * x^1 * y^(n-1) + C(n,n) * x^0 * y^n

where C(n,k) is the binomial coefficient, given by C(n,k) = n! / (k! * (n-k)!).

In our case, the expression (x + xlog10x)^5 has x as the base term and xlog10x as the exponent term. So we can rewrite the expression as:

(x + xlog10x)^5 = x^5 + C(5,1) * x^4 * (xlog10x) + C(5,2) * x^3 * (xlog10x)^2 + ... + C(5,5) * x^0 * (xlog10x)^5

Now let's find the coefficient of the third term. The third term has x^3 as the base term and (xlog10x)^2 as the exponent term. So the coefficient of the third term is given by:

C(5,2) = 5! / (2! * (5-2)!) = 10

So the third term is 10 * x^3 * (xlog10x)^2.

Given that the third term is 106, we have:

10 * x^3 * (xlog10x)^2 = 106

Now we can simplify this equation to find the value of x. Let's break it down step by step:

10 * x^3 * (xlog10x)^2 = 106
x^3 * (xlog10x)^2 = 106 / 10
x^3 * (xlog10x)^2 = 10.6

Taking the logarithm of both sides, we get:

log(x^3 * (xlog10x)^2) = log(10.6)

Using the properties of logarithms, we can simplify the left side:

log(x^3) + log((xlog10x)^2) = log(10.6)
3log(x) + 2log(xlog10x) = log(10.6)

Now, let's solve this equation for x.