Positive number x, y and z satisfy x y z=10⁸¹ and ( ₁₀ x ) ( ₁₀ y z )+ ( ₁₀ y ) ( ₁₀ z )=468 . Find ( ₁₀ x )^2+ ( ₁₀ y )^2+ ( ₁₀ z )^2 .

Step-by-step solution

aligned (a+b+c)^2 & =a^2+b^2+c^2+2(a b+a c+b c) ( ₁₀ x ) ( ₁₀ y z ) & = ( ₁₀ x ) ( ₁₀ y+ ₁₀ z ) & = ₁₀ x ₁₀ y+ ₁₀ x ₁₀ z and x y z & =10⁸¹ implies ₁₀ x+ ₁₀ y+ ₁₀ z=81 aligned and x y z=10⁸¹ implies ₁₀ x+ ₁₀ y+ ₁₀ z=81 Using the property mentioned first, we get aligned & ( ₁₀ x )^2+ ( ₁₀ y )^2+ ( ₁₀ z )^2 & = ( ₁₀ x+ ₁₀ y+ ₁₀ z )^2-2 ( ₁₀ x ₁₀ y+ ₁₀ x ₁₀ z+ ₁₀ y ₁₀ z ) & ( ₁₀ x )^2+ ( ₁₀ y )^2+ ( ₁₀ z )^2 = (81)^2-2 468 & ( ₁₀ x )^2+ ( ₁₀ y )^2+ ( ₁₀ z )^2 =75 aligned