Simplify the following algebraic expressions :
`- m+n - [2m - { m - 3 (m+n) - (2m - n )}] `

Simplify : 2{m - 3(n + bar(m- 2n))}

Add the following expressions: 3n^(2)+5mn-6m^(2), 2m^(2)-3mn -4n^(2), 2mn-3m^(2)-7n^(2)

Simplify each of the following: 1. m-[m+{m+n-2m-(m-2n)}-n] 2. 3x^2z-4y z+3x y-{x^2z-(x^2z-3y z)-4y z-7z}

Simplify: (6m-n)(6m+n)

Simplify each of the products: (m+n/7)^3\ (m-n/7)

Let m ,n are integers with 0ltnltm . A is the point (m ,n) on the Cartesian plane. B is the reflection of A in the line y=x . C is the reflection of B in the y-axis, D is the reflection of C in the x-axis and E is the reflection of D in the y-axis. The area of the pentagon A B C D E is a. 2m(m+n) b. m(m+3n) c. m(2m+3n) d. 2m(m+3n)

Let m ,n are integers with 0

Consider a point A(m,n) , where m and n are positve intergers. B is the reflection of A in the line y=x , C is the reflaction of B in the y axis, D is the reflection of C in the x axis and E is the reflection of D is the y axis. The area of the pentagon ABCDE is a. 2m (m+n) b. m (m+3n) c. m(2m+3n) d. 2m(m+3n)

Let I_(m","n)= int sin^(n)x cos^(m)x dx . Then , we can relate I_(n ","m) with each of the following : (i) I_(n-2","m) " " (ii) I_(n+2","m) (iii) I_(n","m-2) " " (iv) I_(n","m+2) (v) I_(n-2","m+2)" " I_(n+2","m-2) Suppose we want to establish a relation between I_(n","m) and I_(n","m-2) , then we get P(x)=sin^(n+1)x cos^(m-1)x ...(i) In I_(n","m) and I_(n","m-2) the exponent of cos x in m and m-2 respectively, the minimum of the two is m - 2, adding 1 to the minimum we get m-2+1=m-1 . Now, choose the exponent of sin x for m - 1 of cos x in P(x). Similarly, choose the exponent of sin x for P(x)=(nH)sin^(n)x cos^(m)x-(m-1)sin^(n+2) x cos^(m-2)x . Now, differentiating both the sides of Eq. (i), we get =(n+1)sin^(n)x cos^(m)x-(m-1)sin^(n)x(1-cos^(2)x)cos^(m-2)x =(n+1)sin^(n)x cos^(m)x-(m-1)sin^(n)x cos^(m-2)x+(m-1)sin^(n)x cos^(n)x =(n+m)sin^(n)x cos^(m)x-(m-1)sin^(n)x cos^(m-2)x Now, integrating both the sides, we get sin^(n+1)x cos^(m-1)x=(n+m)I_(n","m)-(m-1)I_(n","m-2) Similarly, we can establish the other relations. The relation between I_(4","2) and I_(2","2) is