Question

no. 2 and no.4
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Answer 1

Answer:

(ii) 1/2

(iv) 2

Step-by-step explanation:

(ii)

$\displaystyle \frac{16\times2^{n+1}-4\times 2^n}{16\times2^{n+2}-2\times2^{n+2}}=\frac{16\times2^{n+1}-4\times 2^n}{16\times2^{n+2}-4\times2^{n+1}}\\\\{}\qquad\qquad\qquad\qquad\quad\ \ =\frac{4\times2^n\bigl(4\times2-1\bigr)}{4\times2^n\bigl(4\times2^2-2\bigr)}\\\\{}\qquad\qquad\qquad\qquad\quad\ \ =\frac{4\times2-1}{4\times4-2}\\\\{}\qquad\qquad\qquad\qquad\quad\ \ =\frac7{14}\\\\{}\qquad\qquad\qquad\qquad\quad\ \ =\frac12$

(iv)

$\displaystyle\frac{16^7\times25^5\times81^3}{15^7\times24^5\times80^3}=\frac{(2^4)^7\times(5^2)^5\times(3^4)^3}{(3\times5)^7\times(2^3\times3)^5\times(2^4\times5)^3}\\\\{}\qquad\qquad\qquad\quad=\frac{2^{28}\times5^{10}\times3^{12}}{3^7\times5^7\times2^{15}\times3^5\times2^{12}\times5^3}\\\\{}\qquad\qquad\qquad\quad=\frac{2^{28}\times5^{10}\times3^{12}}{2^{27}\times3^{12}\times5^{10}}\\\\{}\qquad\qquad\qquad\quad=2$