Question

Tính các biểu thức

3 x (2² + 1) x (2⁴ + 1) x (2⁸ + 1) x...x (2⁶⁴ + 1) + 1

8 x (3² + 1) x (3⁴ + 1) x (3⁸ + 1) x...x (3¹²⁸ + 1) + 1

12 x (5² + 1) x (5⁴ + 1) x (5⁸ + 1) x (5¹⁶ + 1)

Answer 1

3. ( 2² + 1 ).( 2⁴ + 1 ).( 2⁸ + 1 )......( 2⁶⁴ + 1 ) + 1

= ( 2² - 1 ).( 2² + 1 ).( 2⁴ + 1 ).( 2⁸ + 1 )....( 2⁶⁴ + 1 ) + 1

= ( 2⁴ - 1 ).( 2⁴ + 1 ).( 2⁸ + 1 )......( 2⁶⁴ + 1 ) + 1

= ( 2⁸ + 1 ).....( 2⁶⁴ + 1 )  + 1

= ( 2⁶⁴ - 1 ).( 2⁶⁴ + 1 ) + 1

=  2¹²⁸ - 1 + 1

= 2¹²⁸

Answer 2

8.( 3² + 1 ).( 3⁴ + 1 ).( 3⁸ + 1 )....( 3¹²⁸ + 1 ) + 1

= ( 3² - 1 ).( 3² + 1 ).( 3⁴ + 1 ).( 3⁸ + 1 )....( 3¹²⁸ + 1 ) + 1

= ( 3⁴ - 1 ).( 3⁴ + 1 ).( 3⁸ + 1 )....( 3¹²⁸ + 1 ) + 1

= ( 3⁸ - 1 ).( 3⁸ + 1 )....( 3¹²⁸ + 1 ) + 1

= ( 3¹⁶ - 1 )......( 3¹²⁸ + 1 ) + 1

= ( 3¹²⁸ - 1 ).( 3¹²⁸ + 1 ) + 1

=  3²⁵⁶ - 1 + 1

= 3²⁵⁶

Answer 3

12.( 5² + 1 ).( 5⁴ + 1 ).( 5⁸ + 1 ).( 5¹⁶ + 1 )

Nhân cả 2 vế với 2

2 = 24.( 5² + 1 ).( 5⁴ + 1 ).( 5⁸ + 1 ).( 5¹⁶ + 1 )

2 = ( 5² - 1 ).( 5⁴ + 1 ).( 5⁸ + 1 ).( 5¹⁶ + 1 )

2 = ( 5⁸ - 1 ).( 5⁸ + 1 ).( 5¹⁶ + 1 )

2 = ( 5¹⁶ - 1 ).( 5¹⁶ + 1 )

2 = 5³² - 1

 = \(\frac{5^{32}-1}{2}\)