"Question 6 Using identities, evaluate. (i) 71^2 (ii) 99^2 (iii) 102^2 (iv) 998^2 (v) (5.2)^2 (vi) 297 × 303 (vii) 78 × 82 (viii) 8.9^2 (ix) 1.05 × 9.5
Class 8 Algebraic Expressions and Identities Page 152"
Answers
An identity is true only for certain values of its variables. An equation is not an identity.
The following are the identities
(a + b)² = a² + 2ab + b²
(a – b)² = a² – 2ab + b²
(a – b)(a + b) = a² – b²
Another useful identity is
(x + a) (x + b) = x² + (a + b) x + ab
If the given expression is the difference of two squares we use the formula
a² –b² = (a+b)(a-b)
• The above four identities are useful in carrying out squares and products of algebraic expressions. They also allow easy alternative methods to calculate products of numbers and so on.
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Solution:
1) 71²
= (70+1)²
= 70²+ 2×70×1+ 1² [(a + b)² = a² + 2ab + b² ]
= 4900 + 140 +1
= 5041
2) 99²
= (100 -1)²
= 100²- 2×100×1 + 1² [(a – b)² = a² – 2ab + b² ]
= 10000 - 200 + 1
= 9801
3) 102²= (100 + 2)²
= 100²+ 2×100 ×2+ 2² [(a + b)² = a² + 2ab + b² ]
= 10000 + 400 + 4
= 10404
4) 998²= (1000 - 2)²
= 1000² - 2×1000×2 + 2² [(a – b)² = a² – 2ab + b² ]
= 1000000 - 4000 + 4
= 996004
5) 5.2²= (5 + 0.2)²
= 5² + 2×5×0.2 + 0.2² [(a + b)² = a² + 2ab + b² ]
= 25 + 2 + 0.04
= 27.04
6) = (300 - 3 )(300 + 3)
= 300²- 3²
= 90000 - 9
= 89991
7) = (80 - 2)(80 + 2)
= 80² - 2² [(a – b)(a + b) = a² – b²]
= 6400 - 4
= 6396
8) 8.9²= (9 - 0.1)²
= 9² - 2×9×0.1 + 0.1² [ [(a – b)² = a² – 2ab + b² ]]
= 81 - 1.8 + 0.01
= 79.21
9)
1.05 × .95 x 10
= (1+ 0.05)(1 - 0.05) x10 [(a – b)(a + b) = a² – b²[
= [1² - 0.05²] x 10
= [1 - 0.0025] x10
= 0.9975 x 10
= 9.975
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Hope this will help you.....
Answer:
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