a. म.स. (HCF) निकालन्नुहोस। \( x^{2}+2 x y+y^{2}-z^{2} x y^{2}+2 y z+z^{2}-x^{2}(3) \) b. सरल गर्नुहोस: (1) \( \left(\frac{a^{x+y}}{a^{z}}\right)^{x-y}\left(\frac{a^{y+z}}{a^{x}}\right)^{y-z}\left(\frac{a^{z+x}}{a^{y}}\right)^{z-x} \)

Simplify the expression by following steps: - step0: Solution: \(\left(\frac{a^{x+y}}{a^{z}}\right)^{x-y}\left(\frac{a^{y+z}}{a^{x}}\right)^{y-z}\left(\frac{a^{z+x}}{a^{y}}\right)^{z-x}\) - step1: Multiply by \(a^{-n}:\) \(\left(\frac{a^{x+y}}{a^{z}}\right)^{x-y}\left(\frac{a^{y+z}}{a^{x}}\right)^{y-z}\left(a^{z+x}\times a^{-y}\right)^{z-x}\) - step2: Multiply by \(a^{-n}:\) \(\left(\frac{a^{x+y}}{a^{z}}\right)^{x-y}\left(a^{y+z}\times a^{-x}\right)^{y-z}\left(a^{z+x}\times a^{-y}\right)^{z-x}\) - step3: Multiply by \(a^{-n}:\) \(\left(a^{x+y}\times a^{-z}\right)^{x-y}\left(a^{y+z}\times a^{-x}\right)^{y-z}\left(a^{z+x}\times a^{-y}\right)^{z-x}\) - step4: Multiply: \(\left(a^{x+y-z}\right)^{x-y}\left(a^{y+z}\times a^{-x}\right)^{y-z}\left(a^{z+x}\times a^{-y}\right)^{z-x}\) - step5: Transform the expression: \(a^{\left(x+y-z\right)\left(x-y\right)}\left(a^{y+z}\times a^{-x}\right)^{y-z}\left(a^{z+x}\times a^{-y}\right)^{z-x}\) - step6: Multiply: \(a^{\left(x+y-z\right)\left(x-y\right)}\left(a^{y+z-x}\right)^{y-z}\left(a^{z+x}\times a^{-y}\right)^{z-x}\) - step7: Transform the expression: \(a^{\left(x+y-z\right)\left(x-y\right)}\times a^{\left(y+z-x\right)\left(y-z\right)}\left(a^{z+x}\times a^{-y}\right)^{z-x}\) - step8: Multiply: \(a^{\left(x+y-z\right)\left(x-y\right)}\times a^{\left(y+z-x\right)\left(y-z\right)}\left(a^{z+x-y}\right)^{z-x}\) - step9: Transform the expression: \(a^{\left(x+y-z\right)\left(x-y\right)}\times a^{\left(y+z-x\right)\left(y-z\right)}\times a^{\left(z+x-y\right)\left(z-x\right)}\) - step10: Multiply the terms: \(a^{\left(x+y-z\right)\left(x-y\right)+\left(y+z-x\right)\left(y-z\right)+\left(z+x-y\right)\left(z-x\right)}\) - step11: Add the terms: \(a^{0}\) - step12: Evaluate the power: \(1\) Let's solve the problems step by step. ### a. Finding the HCF (Highest Common Factor) The expression given is: \[ x^{2}+2xy+y^{2}-z^{2}xy^{2}+2y z+z^{2}-x^{2}(3) \] However, it seems that the expression is not in a standard form for finding the HCF. The HCF is typically calculated for a set of numbers or polynomials. Since the expression is complex and does not represent a clear set of terms, I cannot directly compute the HCF from it. If you have specific terms or polynomials you want to find the HCF for, please provide them. ### b. Simplifying the Expression The expression to simplify is: \[ \left(\frac{a^{x+y}}{a^{z}}\right)^{x-y}\left(\frac{a^{y+z}}{a^{x}}\right)^{y-z}\left(\frac{a^{z+x}}{a^{y}}\right)^{z-x} \] The result of this simplification is: \[ 1 \] This means that the entire expression simplifies to 1. If you have any further questions or need additional assistance, feel free to ask!