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उपयुक्त सर्वसमिका के उपयोंग से निम्नलिखित के मान ज्ञात कीजिए (997)³ (ii) (103)³

Mathematics

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Answers

2 parts
i
991,026,973
ii
1,092,727

Step-by-step solution

6 steps

Here are the values of the given expressions using appropriate identities.

Part i· 3 steps

We need to find the value of (997)3(997)^3.

We can write 997 as (1000−3)(1000 - 3).
We use the identity (a−b)3=a3−3a2b+3ab2−b3(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3.
Here, a=1000a = 1000 and b=3b = 3.

  1. Substitute aa and bb into the identity.

    (997)3=(1000−3)3=(1000)3−3(1000)2(3)+3(1000)(3)2−(3)3(997)^3 = (1000 - 3)^3 = (1000)^3 - 3(1000)^2(3) + 3(1000)(3)^2 - (3)^3

  2. Calculate each term.

    (1000)3=1,000,000,000(1000)^3 = 1,000,000,000 3(1000)2(3)=3(1,000,000)(3)=9,000,0003(1000)^2(3) = 3(1,000,000)(3) = 9,000,000 3(1000)(3)2=3(1000)(9)=27,0003(1000)(3)^2 = 3(1000)(9) = 27,000 (3)3=27(3)^3 = 27

  3. Combine the terms.

    =1,000,000,000−9,000,000+27,000−27= 1,000,000,000 - 9,000,000 + 27,000 - 27 =991,000,000+27,000−27= 991,000,000 + 27,000 - 27 =991,027,000−27= 991,027,000 - 27 =991,026,973= 991,026,973

    The value is:

    991,026,973\boxed{991,026,973}

Part ii1,092,727

We need to find the value of (103)3(103)^3.

We can write 103 as (100+3)(100 + 3).
We use the identity (a+b)3=a3+3a2b+3ab2+b3(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3.
Here, a=100a = 100 and b=3b = 3.

  1. Substitute aa and bb into the identity.

    (103)3=(100+3)3=(100)3+3(100)2(3)+3(100)(3)2+(3)3(103)^3 = (100 + 3)^3 = (100)^3 + 3(100)^2(3) + 3(100)(3)^2 + (3)^3

  2. Calculate each term.

    (100)3=1,000,000(100)^3 = 1,000,000 3(100)2(3)=3(10,000)(3)=90,0003(100)^2(3) = 3(10,000)(3) = 90,000 3(100)(3)2=3(100)(9)=2,7003(100)(3)^2 = 3(100)(9) = 2,700 (3)3=27(3)^3 = 27

  3. Combine the terms.

    =1,000,000+90,000+2,700+27= 1,000,000 + 90,000 + 2,700 + 27 =1,090,000+2,700+27= 1,090,000 + 2,700 + 27 =1,092,700+27= 1,092,700 + 27 =1,092,727= 1,092,727

    The value is:

    1,092,727\boxed{1,092,727}
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