1) 8765 × 974 – 8765 × 874
Explanation:सुत्र:- a × b – a × c = a(b – c)
8765 (974 – 874)
8765 × 100 = 876500
2) 6784 × 786 + 6784 × 214 = ?
Explanation: सुत्र:- a x b + a x c = a(b + c)
6784 (786 + 214)
6784 x 1000 = 6784000
3) 883 x 883 – 117 x 117 = ?
Explanation:सुत्र:- \[a^{2} - b^{2} = (a + b) (a - b)\\ 8832 - 1172 = (883 + 117)(883 - 117)\\ 1000 \times 766 = 766000\\\]
4) (220 x 220 + 180 x 180) = ?
Explanation: सुत्र:- \[a^{2} + b^{2} \: \: \: \: = \frac{1}{2} {(a+b)^2+(a-b)^2 }\\ = \frac{1}{2} {(220+180)^2+(220-180)^2 }\\ = \frac{1}{2} \left \{ 400^2+40^2 \right \}\\ = \frac{1}{2} \times 160000 + 1600\\ = \frac{1}{2} \times 161600 = 80800\]
5) \[\frac{(867 + 289)^2- (867 - 289)^2}{867 \times 289}\]
Explanation: सुत्र:- \[\frac{(a + b)^2- (a - b)^2}{ab } = \frac{4ab}{ab} = 4\]
6) \[\frac{(956+479)^2- (956-479)^2}{(956 \times 956 + 479 \times 479)} = ?\]
Explanation: सुत्र:- \[\frac{(a + b)^2+ (a - b)^2}{a^2+b^2 } =\frac{2(a^2+b^2)}{a^2+b}= 2\]
7) \[\frac{783 \times 783 \times 783 + 217 \times 217 \times 217}{783 \times 783 - 783 \times 217 \times 217 \times 217} = ?\]
Explanation: सुत्र:- \[\frac{(a^3+b^3)}{a^2 -ab+b^2} = a + b\\ a = 783, b = 217\\ 783 + 217 = 1000\]
8) \[\frac{693 \times 693 \times 693- 383 \times 383 \times 383}{693 \times 693+ 693 \times 383+ 383 \times 383}= ?\]
Explanation: सुत्र:- \[\frac{(a^3-b^3)}{a^2 -ab+b^2} = a - b\\ a = 693, b = 383\\ 693 - 383 = 310\]
9) \[\frac{3.4 \times 3.4+ 2 \times 3.4 \times 5.6 \times 5.6}{3.4+5.6} = ?\]
Explanation: सुत्र:- \[\frac{(a^2+2ab+b^2)}{a+b} = \frac{(a+b)^2}{a+b}= a+b\\ a = 3.4, b = 5.6\\ a + b = 3.4 + 5.6 = 9\]
10) \[65 \times 65 + 2\times 65 \times 3 + 3 \times 3\]
Explanation: सुत्र:- \[a^{2} + 2ab + b^{2} = (a+b)^{2}\\ a = 65, b = 3\\ (a + b)^{2} = (65 + 3)^{2} = 68^{2} = 4624\\\]