Simplify \(\left(\displaystyle\frac{x^m}{x^n}\right)^{m+n}\cdot\left(\displaystyle\frac{x^n}{x^p}\right)^{n+p}\cdot\left(x^p\times x^m\right)^{p-m}\)
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 1
Two laws of exponents do the work: \(\displaystyle\frac{x^a}{x^b}=x^{a-b}\) and \((x^a)^b=x^{ab}\). Each bracket becomes a single power of \(x\); the powers are then multiplied by adding their exponents.
The first bracket becomes \(\left(x^{m-n}\right)^{m+n}=x^{(m-n)(m+n)}=x^{m^2-n^2}\)
The second bracket becomes \(\left(x^{n-p}\right)^{n+p}=x^{n^2-p^2}\)
The third bracket becomes \(\left(x^{p+m}\right)^{p-m}=x^{p^2-m^2}\)
Sum of the exponents: \((m^2-n^2)+(n^2-p^2)+(p^2-m^2)=0\)
So the expression is \(x^0=1\)