Dear students, to solve the large expression type approximation problems quickly you must remember few algebraic formulas which we have enlisted below.
1. (a+b)^2 = a^2+b^2+2ab
2. (a-b)^2 = a^2+b^2-2ab
3. (a+b)^2 = (a-b)^2+4ab
4. (a-b)^2 = (a+b)^2-4ab
5. (a+b)^3 = a^3+b^3+3ab(a+b)
6. (a-b)^3 = a^3-b^3-3ab(a-b)
7. a^3+b^3 = (a+b)(a^2-ab+b^2)
8. a^3-b^3 = (a-b)(a^2+ab+b^2)
9. (a+b+c)^2 = a^2+b^2+c^2+2(ab+bc+ca)
10. a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)
Shortcuts Solution of the Questions given below:
Q1. (299.12 + 312.01)^2 / (299.12 + 312.01)
Shortcut Solution: Apply 1st formula:
(299.12 + 312.01)^2 / (299.12 + 312.01)
=(299.12 + 312.01)((299.12 + 312.01)) / (299.12 + 312.01)
=(299.12 + 312.01)
=299+312
=611
Q2. (998.10)^3 + (888.10 )^3 / (998.10)^2+(888.10)^2-(998.10*888.10)
Shortcut Solution: Apply 7th formula:
(998.10)^3 + (888.10 )^3 / (998.10)^2+(888.10)^2-998.10*888.10
=(998.10+888.10 )(998.10)^2+(888.10)^2-998.10*888.10) / (998.10)^2+(888.10)^2-(998.10*888.10)
=998.10+888.10
=998+888
=1886
Q3. (1.33 - 1.22)^2 / (1.33^2 - 1.33*1.22 + 1.22^2)
Shortcut Solution: Apply 2nd formula:
(1.33 - 1.22)^2 / (1.33^2 - 1.33*1.22 + 1.22^2)
= (1.33^2 - 1.33*1.22 + 1.22^2) / (1.33^2 - 1.33*1.22 + 1.22^2)
=1
Q4. (11.11)^3+(22.22)^3+(33.33)^3-3*11.11*22.22*33.33 / (11.11)^2+(22.22)^2+(33.33)^2-11.11*22.22-22.22*33.33-33.33*11.11
Shortcut Solution: Apply 10th formula:
(11.11)^3+(22.22)^3+(33.33)^3-3*11.11*22.22*33.33 / (11.11)^2+(22.22)^2+(33.33)^2-11.11*22.22-22.22*33.33-33.33*11.11
=(11.11+22.22+33.33)(11.11)^2+(22.22)^2+(33.33)^2-11.11*22.22-22.22*33.33-33.33*11.11 / (11.11)^2+(22.22)^2+(33.33)^2-11.11*22.22-22.22*33.33-33.33*11.11
= (11.11+22.22+33.33)
=11+22+33
=66
Q5. (5.14)^3 - (3.12)^3 / (5.14)^2+5.14*3.12+(3.12)^2
Shortcut Solution: Apply 8th formula:
(5.14)^3 - (3.12)^3 / (5.14)^2+5.14*3.12+(3.12)^2
= (5.14 - 3.12)(5.14)^2+5.14*3.12+(3.12)^2 / (5.14)^2+5.14*3.12+(3.12)^2
=5.14-3.12
=5-3
=2
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