Two circles intersect at the points $C$ and $D$. The straight lines $CD$ and $BYXA$ intersect at the point $Z$. Moreever, the straight line $WB$ is tangent to both of the circles. Suppose $ZX=ZY$ and $AB\cdot AX=100$. Find the value of $BW$.
1. **Given Information and Setup:**
   - Two circles \(\Gamma_1\) and \(\Gamma_2\) intersect at points \(C\) and \(D\).
   - A line intersects \(\Gamma_1\) at points \(A\) and \(Y\), intersects segment \(CD\) at point \(Z\), and intersects \(\Gamma_2\) at points \(X\) and \(B\) in the order \(A, X, Z, Y, B\).
   - The line \(WB\) is tangent to both circles at point \(W\) on \(\Gamma_1\).
   - It is given that \(ZX = ZY\) and \(AB \cdot AX = 100\).

2. **Symmetry and Power of a Point:**
   - Since \(ZX = ZY\), point \(Z\) is the midpoint of segment \(XY\). This implies that \(Z\) is the center of the circle passing through \(X\) and \(Y\) with \(Z\) as the midpoint.
   - By the Power of a Point theorem, for point \(A\) with respect to circle \(\Gamma_2\), we have:
     \[
     AB \cdot AX = AZ \cdot AZ = 100
     \]
     This implies that \(AZ^2 = 100\), so \(AZ = \sqrt{100} = 10\).

3. **Tangency Condition:**
   - Since \(WB\) is tangent to both circles at point \(W\) on \(\Gamma_1\), we use the fact that the tangents from a point to a circle are equal. Let \(BW = x\).
   - The tangents from \(W\) to \(\Gamma_1\) and \(\Gamma_2\) are equal, so \(BW = x\).

4. **Using the Given Product:**
   - Given \(AB \cdot AX = 100\), we need to find the value of \(BW\).
   - Since \(W\) is the point of tangency and \(BW\) is the tangent from \(B\) to \(\Gamma_1\), we use the fact that the length of the tangent from an external point to a circle is equal to the square root of the power of the point with respect to the circle.
   - Therefore, \(BW = \sqrt{AB \cdot AX} = \sqrt{100} = 10\).

The final answer is \(\boxed{10}\).