Question

Please solve all the parts using identities
The one who will solve all will be followed, liked and marked as brainlist​

Answer 1

Question : -

Evaluate :

( a ) 1007 x 993 ( b ) 78 x 82 ( c ) 88² - 12²

( d ) 499² ( e ) 103 x 96 ( f ) 256²

( g ) 723² - 23² ( h ) 406 x 394 ( i ) 105² - 99²

Answer : -

Identities used : -

( x + y )² = x² + y² + 2xy

( x - y )² = x² + y² - 2xy

( x + a ) ( x + b ) = x² + x ( a + b ) + ab

( a + b ) ( a - b ) = a² - b²

Solution : -

( a )

1007 x 993

This can be written as ;

( 1000 + 7 ) ( 1000 - 7 )

This in the form of the identity ;

( a + b ) ( a - b ) = a² - b²

This implies ;

( 1000 + 7 ) ( 1000 - 7 ) =

=> ( 1000 )² - ( 7 )²

=> 1000000 - 49

=> 9,99,951

Hence,

( 1000 + 7 ) ( 1000 - 7 ) = 9,99,951

( b )

78 x 82

This can be written as ;

( 80 - 2 ) ( 80 + 2 )

This in the form of the identity ;

( a + b ) ( a - b ) = a² - b²

This implies ;

( 80 - 2 ) ( 80 + 2 ) =

=> ( 80 )² - ( 2 )²

=> 6400 - 4

=> 6396

Hence,

( 80 - 2 ) ( 80 + 2 ) = 6,396

( c )

88² - 12²

This can be written as;

( 80 + 8 )² - ( 10 + 2 )²

This is in the form of an identity ;

( ( 80 )² + ( 8 )² + 2 ( 80 ) ( 8 ) ) - ( ( 10)² + ( 2 )² + 2 ( 10 ) ( 2 ) )

[ 6400 + 64 + 2480] - [ 100 + 4 + 40 ]

8944 - 144

8800

Hence,

( 80 + 8 )² - ( 10 + 2 )² = 8800

( d )

( 499 )²

This can be written as ;

( 500 - 1 )²

Using the identity ;

( a - b )² = a² + b² - 2ab

This implies ;

( 500 )² + ( 1 )² - 2 ( 500 ) ( 1 )

250000 + 1 - 1000

250001 -1000

2,49,001

Hence,

( 500 - 1 )² = 2,49,001

( e )

103 x 96

This can be written as ;

( 100 + 3 ) ( 100 + ( - 4 ) )

Using the identity ;

( x + a ) ( x + b ) = x² + x ( a + b ) + ab

( 100 )² + 100 ( 3 + ( - 4 ) ) + ( 3 ) ( - 4 )

10000 - 1000 - 12

8,988

Hence,

( 100 + 3 ) ( 100 + ( - 4 ) ) = 8,988

( f )

256²

( 250 + 6 )²

Using the identity ;

( a + b )² = a² + b² + 2ab

This implies ;

( 250 )² + ( 6 )² + 2 ( 250 )( 6 )

62,500 + 36 + 3000

65,536

Hence,

( 250 + 6 )² = 65,536

( g )

723² - 23²

( 730 - 7 )² - ( 30 - 7 )²

Using the identity ;

( x - y )² = x² - 2xy + y²

This implies ;

( 730 )² - 2 ( 730 ) ( 7 ) + ( 7 )² - ( 30 )² + ( 7 )² - 2 ( 30 ) ( 7 )

532900 - 10220 + 49 - 900 + 49 - 420

522729 - 529

522200

Hence,

( 730 - 7 )² - ( 30 - 7 )² = 522200

( h )

406 x 394

( 400 + 6 ) ( 394 - 6 )

Using the identity ;

( a + b ) ( a - b ) = a² - b²

This implies ;

( 400 )² - ( 6 )²

160000 - 36

159964

Hence,

( 400 + 6 ) ( 394 - 6 ) = 1,59,964

( i )

105² - 99²

( 100 + 5 )² - ( 100 - 1 )²

Using the Identities ;

( x + y )² = x² + y² + 2xy

( x - y )² = x² + y² - 2xy

This implies ;

( 100 )² + ( 5 )² + 2 ( 100 ) ( 5 ) - (100 )² + ( 1 )² - 2 ( 100 ) ( 1 )

10000 + 25 + 1000 - 10000 + 1 - 200

11025 - 9801

1,224

Hence,

( 100 + 5 )² - ( 100 - 1 )² = 1,224

Hence Solved ! ✓

Answer 2

$\huge\underline\bold\red{Answer}$

a) 1007 × 993

=> identity used (a + b) (a - b) => a² - b²

=> (1000 + 7) (1000 - 7)

=> (1000)² - (7)²

=> 1000000 - 49

=> 999,951

b) 78 × 82

=> identity used (a + b) (a - b) => a² - b²

=> (80 - 2) (80 + 2)

=> (80)² - (2)²

=> 6400 - 4

=> 6,396

c) 88² - 12²

=> identity used => (a + b)² - (a + b)²

=> (80 + 8)² - (10 + 2)²

=> { (80)² + (8)² + 2 × 8 × 80 } - { (10)² + (2)² + 2 × 10 × 2 }

=> { 6400 + 64 + 2480 } - { 100 + 4 + 40 }

=> 8944 - 144

=> 8,800

d) 499²

=> identity used => (a - b)² => (a + b) (a - b)

=> (500 - 1)²

=> (500)² + (1)² - 2 × 500 × 1

=> 250000 + 1 - 1000

=> 249,001

e) 103 × 96

=> Identity used => (x + a) (x - b) => (x)² + x(a + b) + ab

=> (100 + 3) {100 + (-4) }

=> (100)² + 100{ 3 + (-4) + (3) (-4) }

=> 10000 - 1000 - 12

=> 8,988

f) 256²

=> identity used => (a + b)² => (a)² + (b)² + 2ab

=> (250 + 6)²

=> (250)² + (6)² + 2 × 250 × 6

=> 62500 + 36 + 3000

=> 65,536

g) 723² - 23²

=> identity used => (a - b)² => a² + b² - 2ab

=> (730 - 7)² - (30 - 7)²

=> { (730)² + (7)² - 2 × 730 × 7 } - { (30)² + (7)² - 2 × 30 × 7 }

=> 532900 - 10220 + 49 + 900 + 49 - 420

=> 522729 - 529

=> 522,200

h) 406 × 394

=> identity used => a² - b² => (a + b) (a - b)

=> (400 + 6) (400 - 6)

=> (400)² - (6)²

=> 160000 - 36

=> 159,964

i) 105² - 99²

=> Identity used => (a - b)² , (a + b)2²

=> (100 + 5)² - (100 - 1)²

=> { (100)² + (5)² + 2 × 100 × 5 } - { (100)² + (1)² - 2 × 100 × 1 }

=> 10000 + 25 + 1000 - 10000 + 1 - 200

=> 11025 - 9801

=> 1,224

_____________________