If x+y+z=7, x^2+y^2+z^2=85 and x^3+y^3+z^3=913, then value of 3√xyz is Option 1: 1 Option 2: 2 Option 3: 4 Option 4: 8

Answer: 4. Solution: Using the identity (x+y+z)^2=x^2+y^2+z^2+2(xy+yz+zx): 7^2=85+2(xy+yz+zx) 49-85=2(xy+yz+zx) xy+yz+zx=-18 Now using the identity x^3+y^3+z^3-3xyz=(x+y+z)(x^2+y^2+z^2-xy-yz-zx): 913-3xyz=7(85-(-18)) 913-3xyz=7x103=721 3xyz=192 xyz=64 3√xyz=3√64=4

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If $x+y+z=7$ , $x^2+y^2+z^2=85$ and $x^3+y^3+z^3=913$ , then value of $\sqrt[3]{xyz}$ is

✅ Correct Option: 3

Using the identity $(x+y+z)^2=x^2+y^2+z^2+2(xy+yz+zx)$ :

$7^2=85+2(xy+yz+zx)$

$\Rightarrow 49-85=2(xy+yz+zx)$

$\Rightarrow xy+yz+zx=-18$

Now using the identity $x^3+y^3+z^3-3xyz=(x+y+z)\left(x^2+y^2+z^2-xy-yz-zx\right)$ :

$913-3xyz=7\left(85-(-18)\right)$

$\Rightarrow 913-3xyz=7\times103=721$

$\Rightarrow 3xyz=192$

$\Rightarrow xyz=64$

$\sqrt[3]{xyz}=\sqrt[3]{64}=4$

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If x+y+z=7x+y+z=7x+y+z=7, x2+y2+z2=85x^2+y^2+z^2=85x2+y2+z2=85 and x3+y3+z3=913x^3+y^3+z^3=913x3+y3+z3=913, then value of xyz3\sqrt[3]{xyz}3xyz​ is

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If $x+y+z=7$ , $x^2+y^2+z^2=85$ and $x^3+y^3+z^3=913$ , then value of $\sqrt[3]{xyz}$ is