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 the problem, we need to simplify the expression $\sqrt{21 - 4\sqrt{5} + 8\sqrt{3} - 4\sqrt{15}}$ and then find the values of $x$, $y$, and $z$ such that $\sqrt{x} + \sqrt{y} - \sqrt{z}$ equals this simplified expression. Finally, we will calculate $xyz$.

Step 1: Simplify the Expression

We start with the expression inside the square root: $21 - 4\sqrt{5} + 8\sqrt{3} - 4\sqrt{15}$

We need to check if this can be expressed as a square of a binomial of the form $(\sqrt{a} + \sqrt{b} - \sqrt{c})^2$.

Step 2: Expand the Binomial

Consider the binomial $(\sqrt{a} + \sqrt{b} - \sqrt{c})^2$: $(\sqrt{a} + \sqrt{b} - \sqrt{c})^2 = a + b + c + 2\sqrt{ab} - 2\sqrt{ac} - 2\sqrt{bc}$

Step 3: Match Terms

We need to match the terms from the expanded binomial to the given expression: $a + b + c = 21$ $2\sqrt{ab} = 8\sqrt{3}$ $-2\sqrt{ac} = -4\sqrt{5}$ $-2\sqrt{bc} = -4\sqrt{15}$

Step 4: Solve for $a$, $b$, and $c$

From $2\sqrt{ab} = 8\sqrt{3}$: $\sqrt{ab} = 4\sqrt{3}$ $ab = 48$

From $-2\sqrt{ac} = -4\sqrt{5}$: $\sqrt{ac} = 2\sqrt{5}$ $ac = 20$

From $-2\sqrt{bc} = -4\sqrt{15}$: $\sqrt{bc} = 2\sqrt{15}$ $bc = 60$

Step 5: Solve the System of Equations

We have the following system of equations: $a + b + c = 21$ $ab = 48$ $ac = 20$ $bc = 60$

To find $a$, $b$, and $c$, we can use the relationships between the products and sums of the roots.

Step 6: Find $a$, $b$, and $c$

From $ab = 48$ and $ac = 20$: $b = \frac{48}{a}$ $c = \frac{20}{a}$

Substitute these into $bc = 60$: $\left(\frac{48}{a}\right) \left(\frac{20}{a}\right) = 60$ $\frac{960}{a^2} = 60$ $a^2 = \frac{960}{60}$ $a^2 = 16$ $a = 4$

Now, find $b$ and $c$: $b = \frac{48}{4} = 12$ $c = \frac{20}{4} = 5$

Step 7: Verify the Values

Check if $a + b + c = 21$: $4 + 12 + 5 = 21$

Step 8: Calculate $xyz$

Given $x = 4$, $y = 12$, and $z = 5$: $xyz = 4 \times 12 \times 5 = 240$

Final Answer

The value of $xyz$ is $\boxed{240}$.