Final answer:
The expression simplifies using exponent rules, including the power of a power rule and the quotient of powers rule, resulting in the simplified form X^(a+b+c), which coincides with option (b).
Step-by-step explanation:
The student is asking to simplify the expression (Xᵅ/x.x⁻ᵈ)ᵅ-ᵈ × (Xᵇ/x.x⁻c)ᵇ-ᵈ × (xᵌ/x.x⁻a)ᵌ-ᵊ. This expression can be simplified using exponent rules, specifically the power of a power rule and the quotient of powers rule. These rules state that (xʸ)ᵇ = xʸ.ᵇ and xʸ/xʹ = xʸ-ʹ. Applying these rules to each part of the expression:
- (Xᵅ/x.x⁻ᵈ) becomes Xᵅ-(-ᵈ), which simplifies to Xᵅ+ᵈ.
- (Xᵇ/x.x⁻c) becomes Xᵇ-(-ᵈ), which simplifies to Xᵇ+ᵈ.
- (xᵌ/x.x⁻a) becomes xᵌ-(-ᵊ), which simplifies to xᵌ+ᵊ.
Now, combining all parts:(Xᵅ+ᵈ) × (Xᵇ+ᵈ) × (xᵌ+ᵊ), we get Xᵅ+ᵇ+ᵌ. Therefore, the simplified expression is Xᵅ+ᵇ+ᵌ, which matches option (b).