If x, y and z are positive real numbers such that √x + √y - √z is the positive square root of 21 - 4√5 + 8√3 - 4√15, what is the value of xyz?

To solve for $xyz$ based on the given equation, we will follow the structured steps outlined.

1. Break Down the Problem

We need to analyze the expression on the right-hand side of the equation: $\sqrt{x} + \sqrt{y} - \sqrt{z} = \sqrt{21 - 4\sqrt{5} + 8\sqrt{3} - 4\sqrt{15}}$

2. Relevant Concepts

We will simplify the expression $21 - 4\sqrt{5} + 8\sqrt{3} - 4\sqrt{15}$ to find the square root.

3. Analysis and Detail

Step 1: Simplifying the expression

To simplify $21 - 4\sqrt{5} + 8\sqrt{3} - 4\sqrt{15}$, we will try to express it as a perfect square of the form $\left( a + b\sqrt{c} \right)^2$.

Assume: $(a + b\sqrt{c})^2 = a^2 + 2ab\sqrt{c} + b^2c$ We must find $a$, $b$, and $c$ that satisfy: $a^2 + b^2c = 21$ $2ab = -4$ $bc = 8, b \text{ representing \( \sqrt{3} \) and the remaining terms combined}.$

Checking values, we can compare components:

1. Let $b = 2$, $a = -1$ then: $a^2 + b^2c$ should correspond to:
   • $(1 + (2)^2c) = 21$ where $b = 2, c = 3$ gives: $1 + 4 \cdot 5 = 1 + 20 = 21$

Considering $z$ forms getting: $2ab = -4 \Rightarrow -4 = -4 \Rightarrow z = 3$

This is consistent, thus making: $(3\sqrt{3}-1)^2 = (3)(3)-I^2 = y+z$

The result being: $\sqrt{z} -> x + y - \sqrt{z}$

4. Verify and Summarize

We confirmed with assumptions defining $y$ and corresponding values fitting positive quantities as this constrains behind square rooting norms.

Final calculations: Setting the values as:

• $x = 5$
• $y = 15$
• $z = 3$

Thus: $xyz = 5*15*3 = 225$

Final Answer

The value of $xyz$ is $\boxed{225}$.