With the pocket calculator:
123456789012345678901234567890 + 678901234567890123456789012345 = 8.023580235802358e+29
With this I have the exact value:
123456789012345678901234567890 + 678901234567890123456789012345 = 802358023580235802358023580235

With the pocket calculator:   123456789123456789 * 8001 = 987777769776777900000
With this I have the exact value:
123456789123456789 x 8001 = 987777769776777768789

With the pocket calculator:   123456789123456789 ^ 3 = 1.8816763774341882e+51
With this I have the exact value:
123456789123456789 x 123456789123456789 = 15241578780673678515622620750190521
123456789123456789 x 15241578780673678515622620750190521 = 1881676377434183981909562699940347954480361860897069

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x*(10+x)+5*x^3 where x = 123456789012345:  9408381861768148891412848380186556157340600
Indeed:

10+x
123456789012345 + 10 = 123456789012355

x*(10+x)
123456789012355 x 123456789012345 = 15241578753239903688452522475

x^2
123456789012345 x 123456789012345 = 15241578753238669120562399025

x^3
15241578753238669120562399025 x 123456789012345 = 1881676372353626729966819028056573540963625

5*x^3
1881676372353626729966819028056573540963625 x 5 = 9408381861768133649834095140282867704818125

x*(10+x)+5*x^3
9408381861768133649834095140282867704818125 + 15241578753239903688452522475 = 9408381861768148891412848380186556157340600

With the pocket calculator:   9.40838186176815e+42

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30! = 265252859812191058636308480000000
I can get 30! directly with factorial 2. In any case:
2 x 1 = 2
3 x 2 = 6
4 x 6 = 24
...
25 x 620448401733239439360000 = 15511210043330985984000000
26 x 15511210043330985984000000 = 403291461126605635584000000
27 x 403291461126605635584000000 = 10888869450418352160768000000
28 x 10888869450418352160768000000 = 304888344611713860501504000000
29 x 304888344611713860501504000000 = 8841761993739701954543616000000
30 x 8841761993739701954543616000000 = 265252859812191058636308480000000

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Euler conjectured that there were no positive integers x, y, z and w such that x^4 = y^4 + z^4 + w^4.
In 1988 this conjecture was shown to be false by taking x = 422481, y = 95800, z = 217519 and w = 414560:

31774520764038345920321 + 84229075969600000000 = 31858749840007945920321
29535857400192040960000 + 2238663363846304960321 = 31774520764038345920321
[ 95800^4+217519^4+414560^4 / ]

171859993600 x 171859993600 = 29535857400192040960000
414560 x 414560 = 171859993600
[ 414560^4 / ]

47314515361 x 47314515361 = 2238663363846304960321
217519 x 217519 = 47314515361
[ 217519^4 / ]

9177640000 x 9177640000 = 84229075969600000000
95800 x 95800 = 9177640000
[ 95800^4 / ]

178490195361 x 178490195361 = 31858749840007945920321
422481 x 422481 = 178490195361
[ 422481^4 / ]

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1881676377434183981909562699940347954480361860897069 / 5 ?
/ 5  -->  x 2 / 10
1881676377434183981909562699940347954480361860897069 x 2 = 3763352754868367963819125399880695908960723721794138
376335275486836796381912539988069590896072372179413.8

1881676377434183981909562699940347954480361860897069 / 4 ?
/ 4  -->  x 25 / 100
1881676377434183981909562699940347954480361860897069 x 25 = 47041909435854599547739067498508698862009046522426725
470419094358545995477390674985086988620090465224267.25

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If I know that √2 = 1.414213562373095048801688724209698..., how can I find three more digits of √2?
14142135623730950488016887242096981 x 14142135623730950488016887242096981 = 200000000000000000000000000000000006061412135989499098047242209314361
So:  √2 = 1.4142135623730950488016887242096980...
141421356237309504880168872420969808 x 141421356237309504880168872420969808 = 20000000000000000000000000000000000040455788649711890284048731247556864
141421356237309504880168872420969807 x 141421356237309504880168872420969807 = 19999999999999999999999999999999999757613076175092880523710986405617249
So:  √2 = 1.41421356237309504880168872420969807...
1414213562373095048801688724209698078 x 1414213562373095048801688724209698078 = 1999999999999999999999999999999999998388724615478808833198118227916894084
1414213562373095048801688724209698079 x 1414213562373095048801688724209698079 = 2000000000000000000000000000000000001217151740224998930801495676336290241
So:  √2 = 1.414213562373095048801688724209698078...