Find the value of X power 3 Y power 3 Z power three minus 3XY zee if X square plus Y Square plus Z Square is equal to 83 and X plus Y plus Z is equal to 15?

Question

Find the value of X power 3  Y power 3  Z power three minus 3XY zee if X square plus Y Square plus Z Square is equal to 83 and X plus Y plus Z is equal to 15?

Answer

Finding the Value of X³Y³Z³ - 3XYZ Given X² + Y² + Z² = 83 and X + Y + Z = 15

Step 1: Simplifying the expression

We can start by simplifying the expression X³Y³Z³ - 3XYZ using algebraic identities. We know that:

(X + Y + Z)³ = X³ + Y³ + Z³ + 3(X²Y + XY² + X²Z + XZ² + Y²Z + YZ²) + 6XYZ

(X + Y)² = X² + Y² + 2XY

(X - Y)² = X² + Y² - 2XY

Using the first identity, we can rewrite X³Y³Z³ as:

X³Y³Z³ = (X + Y + Z)³ - 3(X²Y + XY² + X²Z + XZ² + Y²Z + YZ²) - 6XYZ

Substituting X + Y + Z = 15 and simplifying the expression, we get:

X³Y³Z³ = 3375 - 3(XY(Z + Y) + XZ(Y + Z) + YZ(X + Z)) - 6XYZ

Now, we can simplify the expression 3XY(Z + Y) + XZ(Y + Z) + YZ(X + Z) using the second and third identities:

3XY(Z + Y) + XZ(Y + Z) + YZ(X + Z) = (3XYZ + XZY + YZX + XYZ) + (3XY² + 3XZ² + 3YZ² - 2XYZ)

Substituting this expression in the previous equation, we get:

X³Y³Z³ = 3375 - 3(3XYZ + XZY + YZX + XYZ + 3XY² + 3XZ² + 3YZ² - 2XYZ) - 6XYZ

Expanding and simplifying the expression further, we get:

X³Y³Z³ - 3XYZ = 3375 - 9XYZ(X + Y + Z) - 3(X²Y² + X²Z² + Y²Z²) + 6XYZ

X³Y³Z³ - 3XYZ = 3375 - 9XYZ(15) - 3(X²Y² + X²Z² + Y²Z²)

X³Y³Z³ - 3XYZ = 3375 - 135XYZ - 3(X²Y² + X²Z² + Y²Z²)

Step 2: Finding X²Y² + X²Z² + Y²Z²

Next, we need to find the value of X²Y² + X²Z² + Y²Z². We know that:

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