Prove the quotient rule using `a^m/a^n=a^(m-n)`

Prove the rule of exponents (ab)^(n) = a^(n)b^(n) by using principle of mathematical induction for every natural number.

If the straigjht line x cos alpha+ y sin alpha=p touches the curve x^(m)y^(n)=a^(m+n) , prove that. p^(m+n)""m^(m)n^(n)=(m+n)^(m+n)a^(m+n)sin^(n) alpha cos^(m) alpha

Prove : int sin mx sin n x dx[ m^(2) != n^(2)] , = 1/2 [ (sin(m-n)x)/(m-n) - (sin (m+n)x)/(m+n) ] + c

If the line x cos alpha+y sin alpha= p touches the curve x^my^n= a^(m+n) prove that p^(m+n).m^m.n^n= a^(m+n)(m+n)^(m+n).cos^m alpha. Sin^n alpha .

In the expansion of (1+x)^(m+n) , where m & n are +ve integers, prove that the coefficients of x^m and x^n are equal.

Prove that by using the principle of mathematical induction for all n in N : n(n+1)(n+5) is a multiple of 3

Prove that : ((x^m)/(x^n))^(m+n-l) xx((x^n)/(x^l))^(n+l-m)xx((x^l)/(x^m))^(l+m-n) =1 .

If A.M. and G.M. between two numbers is in the ratio m : n then prove that the numbers are in the ratio (m+sqrt(m^2-n^2)):(m-sqrt(m^2-n^2))dot

Prove that : ((x^m)/(x^n))^(m+n) xx((x^n)/(x^l))^(n+l)xx((x^l)/(x^m))^(l+m) =1 .

If the ratio of the roots of the quadratic equation ax^(2)+bx+c=0 be m:n, then prove that ((m+n)^2)/(mn)=(b^(2))/(ac)