Question

Using binomial theorem, find the value of:
i) (9.9)³
ii) (0.9)⁴

Answer 1

Solution :

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Binomial theorem :

Let n be a positive integer and

x , a be real numbers. then

(x+a)ⁿ = ⁿC0xⁿa^0 +ⁿC1x^n-1a¹ +..

...+ⁿCnx^0aⁿ

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Here ,

i ) ( 9.9 )³

= ( 10 - 0.1 )³

= C0(10)³-C1(10)²(0.1)+C2(10)¹(0.1)²-C3(0.1)³

= 1000 - 3×100×0.1+3×10×(0.01)-(0.001)

= 1000 - 30 + 0.3 - 0.001

= 970.299

Therefore ,

( 9.9 )³ = 970.299

ii ) ( 0.9 )⁴

= ( 1 - 0.1 )⁴

= C01⁴-C1( 1 )³(0.1)+C2(1)²(0.1)²

-C3(1)(0.1)³+C4(0.1)⁴

= 1 - 4 × 0.1 + 6 × 0.01 - 4×0.001 + 0.0001

= 1 - 0.4 + 0.06 - 0.004 + 0.0001

= 1.0601 - 0.404

= 0.6561

Therefore ,

( 0.9 )⁴ = 0.6561

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