Answer:
( i ) [tex]\boxed{1}[/tex]
( ii ) [tex]\boxed{2^{6n - 3m - 2}\times 3^{3 - m}}[/tex]
.
Step-by-step explanation:
( Part i )
[tex]\left(\frac{1}{x^{a - b}} \right)^{a + b}\left(\frac{1}{x^{b + a}} \right)^{b - a}[/tex]
[tex]= \left(\frac{1}{x^{(a - b)(a + b)}} \right)\left(\frac{1}{x^{(b + a)(b - a)}} \right)[/tex]
[tex]= \left(\frac{1}{x^{a^2 - b^2}} \right)\left(\frac{1}{x^{b^2-a^2}} \right)[/tex]
[tex]= \frac{1}{x^{a^2 - b^2 + (b^2 - a^2)}}[/tex]
[tex]= \frac{1}{x^0}[/tex]
[tex]= \frac{1}{1}[/tex]
[tex]= \boxed{1}[/tex]
.
( Part ii )
[tex]\frac{2^{m + n}\times 3^{2m + 3} \times 4^{2n -1}}{6^{2m + n}\times 12^{m - n}}[/tex]
[tex]=\frac{2^{m + n}\times 3^{2m + 3} \times 2^{2(2n -1)}}{(2\times3)^{2m + n}\times (3\times 2^2)^{m - n}}[/tex]
[tex]=\frac{2^{m + n}\times 3^{2m + 3} \times 2^{4n -2}}{2^{2m + n}\times3^{2m + n}\times 3^{m - n}\times 2^{2(m - n)}}[/tex]
[tex]=\frac{2^{m + n + 4n - 2}\times 3^{2m + 3}}{2^{2m + n + 2m - 2n}\times3^{2m + n + m - n}}[/tex]
[tex]=\frac{2^{m + 5n - 2}\times 3^{2m + 3}}{2^{4m - n}\times3^{3m}}[/tex]
[tex]= 2^{m + 5n - 2 - (4m - n)}\times3^{2m + 3 - 3m}[/tex]
[tex]= \boxed{2^{6n - 3m - 2}\times 3^{3 - m}}[/tex]
.
Happy to help :)
