(b - 7)² = (b)² - 2(b)(7) + (7)² = b² -14b + 49
[(a - b)² = a² - 2ab + b², here a = b, b = 7]

(a + 3b)\(\times\)(x + 5)
= a\(\times\)(x + 5) + 3b\(\times\)(x + 5)
= (a\(\times\)x) + (a\(\times\)5) + (3b\(\times\)x) + (3b\(\times\)5)
= ax + 5a + 3bx + 15b

a⁴b-a⁶b

We have, 2, 4y, 8y², 16y³
Required product = 2 x (4y) x (8y²) x (16y³)
= (8y) x (8y²) x (16y³)
= (8 x 8) x (y x y²) x (16y³)
= (64y³) x (16y³)
= (64 x 16) x (y³x y³)
=1024y⁶

We have,length = 20x² and breadth = 5y²
∴ Area of rectangle  = LengthｘBreadth
= (20x²) x (5y²) = (20 x 5) x (x² x y²) =100x²y²

4x ｘ 5yｘ 7z=(4xｘ5y)ｘ7z=20xyｘ7z=140 xyz

We have,a-b + ab,b-c + bc,c-a + ac
On writing the given expressions in separate rows with like terms one below the other, we get
a-b + ab
   +b     -c+bc
+-a        +c    +ac
________________
0 + 0 + ab + 0 + bc + ac
________________
Thus, the sum of the expression is ab + be + ca.

We have, (2x + 1)
For given expression (2x + 1), the variable x has a position Xl on the number line. The position of 2x will be a point A, the distance of A from the origin will be two times the distance of Xl from the origin and the value of 2x + 1 is given by a point B, 1 unit to the right of 2x.

(7x² - 4x + 5)+(-3x² + 2x-1) + (5x² -x + 9)
=7x² - 3x² +5x² -4x + 2x - x + 5 - 1 + 9 =(7 - 3 + 5) x² + (-4 + 2 - 1)x + (5 - 1 + 9)
=9x² - 3x + 13

We have, (z - 3)
For given expression (z - 3), the variable z has a position Z on the number line. Z may be anywhere on the number line but the value of(z - 3) is given by position P, 3 units to the left of Z.

= (ab)² + 2(ab)(bc) + (bc)² - 2ab²c
= a²b² + 2ab²c +b²c² - 2ab²c
= a²b² + (2 - 2)ab²c+ b²c²
= a²b² + (0)ab²c + b²c²
= a²b² + 0 + b²c² = a²b² + b²c²

-a²-3b²

14xy,21xy

Given,51²-49²=100p
2601-2401=100p
200 =100p
\(p=\frac { 200 }{ 100 } =2\)
So, it is true
Alternate Method
51²- 49² =100p
(51- 49)(51 + 49) =100p
2 x100 =100p
200 =100p
\(p=\frac { 200 }{ 100 } =2\)

(i)(48)²=(50-2)²
Since, (a - b)² = a² - 2ab + b²
(50-2)²=(50)²-2x50x2+(2)²
= 2500 - 200 + 4   (a-b)2=a²-2ab+b²
= 2504 - 200 = 2304
(ii) 181² -19² =(181-19)(181+ 19)
where, a = 50 and b = 2
= 162 x 200 = 32400
(iii) 497 x 505 =(500 - 3)(500 + 5)
= 5002 + (-3 + 5) x 500 + (-3)(5)
[.: (x + a)(x + b) = x² + (a +. b) x + ab]
= 250000 + 1000 -15 = 250985
(iv) 2.07 x1.93 = (2 + 0.07)(2 - 0.07)
= 2² -(0.07)²
[.: where, a = 50 and b = 2]
= 3.9951

(48)²=(50-2)²
Since, (a - b)² = a² - 2ab + b²
(50-2)²=(50)²-2x50x2+(2)²
= 2500 - 200 + 4
[(a-b)²=a²-2ab+b²]
= 2504 - 200 = 2304
(ii) 181² -19² =(181-19)(181 + 19)
where, a = 50 and b = 2
= 162 x 200 = 32400
(iii) 497 x 505 =(500 - 3)(500 + 5)
= 500² + (-3 + 5) x 500 + (-3)(5)
[.: (x + a)(x + b) = x² + (a +b) x + ab]
= 250000 + 1000 -15 = 250985
(iv) 2.07 x1.93 = (2 + 0.07)(2 - 0.07)
= 2² -(0.07)²
[.: where, a = 50 and b = 2]
= 3.9951

Monomial, binomial, trinomial