Algebraic Expressions And Identities

(i) 7x²yz – 5xy

This equation consists of two terms that are:

7x²yz and - 5xy

The coefficient of 7x²yz is 7

The coefficient of – 5xy is – 5

(ii)

This equation consists of three terms that are:

x², x, 1

The coefficient of x² is 1

The coefficient of x is 1

The coefficient of 1 is 1

(iii)

This equation consists of three terms that are:

3x²y, -5x²y²z² and z²

The coefficient of 3x²y is 3

The coefficient of -5x²y²z² is -5

The coefficient of z² is 1

(iv)

(v)

(vi)

(i) x+y

This expression contains two terms x and y

So, it is called ‘Binomial’

(ii) 1000

It contains one term 1000

So, it is called monomial

(iii) x+x²+x³+x⁴

It contains four terms

So, it is not a monomial, binomial and trinomial

(iv)7+a+5b

It contains three terms

So, it is called trinomial

(v) 2b-3b²

It contains two terms

So, it is called binomial

(vi) 2y-3y²+4y³

It contains three terms

So, it is called trinomial

(vii) 5x-4y+3x

8x – 4y

It contains two terms

So, it is called binomial

(viii) 4a-15a²

It contains two terms

So, it is called binomial

(ix) xy+yz+zt+tx

It contains four terms

So, it is not a monomial, binomial and trinomial

(x) pqr

It contains one term

So, it is called monomial

(xi) p²q+pq²

It contains two terms

So, it is called binomial

(xii) 2p+2q

It contains two terms

So, it is called monomial

(i) 3a²b, -4a²b, 9a²b

= 3a²b + (-4a²b) + 9a²b

= 3a²b – 4a²b + 9a²b

= 3a²b

(ii)

= a + a - a

Taking L.C.M 3, 5 , 5 is 15

=11

=

=

(iii)

= 4xy² – 7x²y + 12x²y – 6xy² – 3x²y + 5xy²

= 4x² + 12x²y – 3x²y – 7x²y – 6xy² + 5xy²

= 3xy² + 2x²y

(iv)

Adding all, we get

=

= + +

= - +

(v)

Adding all, we get

= xy + y + + y - - xy

= + +

= - -

(vi)

Adding all, we get

= x³ - x² + + x³ + x² – x + + x² - x – 2

= x³ + x² - +

= 5x³ + x² -

(i) -5xy from 12xy

After subtracting,we get

= 12xy - (- 5xy)

= 5xy + 12xy

= 17xy

(ii) 2a² from -7a²

After subtracting, we get

= 2a² + (-7a²)

= -2a² + 7a²

= -9a²

(iii) 2a-b from 3a-5b

After subtracting, we get

= -(2a – b)+ (3a – 5b)

= -2a + b+ 3a – 5b

= a – 4b

(iv)

After subtracting, we get

= - (2x³ – 4x² + 3x + 5) + (4x³ + x² + x + 6)

= - 2x³ + 4x² – 3x – 5 + 4x³ + x² + x + 6

= 2x³ + 5x² – 2x + 1

(v)

After subtracting, we get

= y² + y² + y – 2 - y³ + y² + 5

= y³ + y² + y + 3

= y³ + y² + y + 3

(vi)

After subtracting, we get

= x + y - z – (x - y - z)

= x - x + y + y - z + z

= + +

= + +

(vii) x² - xy² + xy from x²y + xy² - xy

= x²y + xy² - xy – (x² - xy² + xy)

= x²y – x²y + xy² + xy² - xy - xy

= x²y + xy² - xy

(viii)

After subtracting, we get

= bc - ac – ( - bc + ac)

= bc + bc - ac - ac -

= + -

= + -

= - -

= – 2ac -

(i)

= x³ - x² + x + – (x² - x³ + + x)

= x³ + x³ - x² - x² + x - x + -

= x³ - x² - -

= x³ - x² - -

(ii)

= a³ – a² – – (a² + a² + –

= a⁵ - a³ - a² - a² - - +

= (2a³ – 9a³) – (3a² – 10a²) – +

= a³ - a² - -

(iii)

= - - x² – (x³ + x² + x + )

= x³ - x² - x² - - + -

= x³ - x²- +

= x³ - x² - – 1

(iv)

= - y² – (y³ + y² + y + )

= y³ - y² - y² - + -

= y³ + (-5y² – 7y²) - y +

= y³ - y² - y -

(v)

= ab - ac - bc – (ac - ab + )

= ab - ab - ac - ac - bc - bc

= - -

= ab - ac - bc

The sum of x – 3y + 2z and -4x + 9y – 11z is calculated as below:

= (x – 3y + 2z) + (-4x + 9y – 11z)

= x – 4x – 3y + 9y + 2z – 11z

= -3x + 6y -9z

Now, The expression 3x- 4y -7z has to be subtracted from the resultant expression i.e. -3x + 6y -9z

= (-3x + 6y -9z) – (3x – 4y – 7z)

= -3x – 3x + 6y + 4y – 9z + 7z

= -6x + 10y – 2z

Subtract the sum of 3l-4m-7n² and 2l+3m-4n² from the sum of 9l+2m-3n² and -3l+m+4n²………

Sum of 9l + 2m – 3n² and -3l + n + 4n²

= 9l + 2m – 3n² + (-3l + m + 4n²)

= 9l – 3l + 2m + m – 3n² + 4n²

= 6l + 3m + n² (i)

Sum of 3l – 4m – 7n² and 2l + 5m – 4n²

= 3l – 4m – 7n² + 2l + 5m – 4n²

= 5l – m – 11n² (ii)

Subtract (i) and (ii), we get

= 6l + 3m + n² – (5l – m – 11n²)

= 6l – 5l + 3m + m + n² + 12n²

= l + 4m + 13n²

As given in the question, the Sum of 2x – x² + 5 and -4x – 3 + 7x² is given as:

= 2x – x² + 5 – 4x – 3 + 7x²

= 2x – 4x – x² + 7x² + 5 – 3

= -2x + 6x² + 2 (i)

Now subtracting equation (i) from 5 we get,

Subtracting (ii) from (i), we get

= 5 - (-2x + 6x² + 2)

= 5 + 2x – 6x² – 2

= 3 + 2x – 6x²

Therefore, the resultant expression is 3 + 2x – 6x²

(i)

= x² - 3x² – 3x + 5x + 5 - 7

= (2x² – 3x²) - (6x + 5x) +

= x² - +

= x² - x +

(ii)

= 5 – 3x + 2y – 2x + y – 3x + 7y – 9

= - 8x + 10y – 4

(iii)

= x²y + x²y - xy² - xy² + xy + xy

= (165x²y + 2x²y) + (-126xy² – 4xy²) +

= x²y - xy² + xy

= x²y - xy² + xy

(iv)

= y² – 2y² - y² - y - y - y + 11 + 3 – 2

= (y² – 6y² + 2y²) - (4y – y – 2y) + 14 – 2

= y² - y + 12

= -y² – y + 12

(v)

= a²b²c - a²b²c + ab²c + ab²c - abc² - abc² + a²bc

= a²b²c + ab²c - abc² + a²bc

5 × x × x × 4 × x × x × x

= 5 × 4 × x⁵

= 20 × x⁵

= 20x⁵

- 3 × 4 – a² × b²

= -12 × a² × b²

= -12a²b²

(-5) × (-5) × x × x² × y × y × z

= 15 × x³ × y² × z

= 15x³y²z

× × x × x² × y × y × z²

= × x³ × y² × z²

= x³y²z²

× × x × x² × y² × y × z × z²

= × x³ × y³ × z³

= x³y³z³

× × x³ × x × z × z² × y

= × x⁴ × z³ × y

= x⁴z³y

× × a² × a³ × b² × b² × c²

= x a⁵ × b⁴ × c²

= a⁵b⁴c²

-7 × × x × y × x² × y × z

= × x3 × y² × z

= x³y²z

7 × -5 × 6 × a × a × a × b × b² × b × c × c²

= 210 × a³ × b⁴ × c³

= 210a³b⁴c³

(-5) × (-10) × (-2) × a × a² × a³

= -100 × a⁶

= -100a⁶

(-4) × (-6) – (-3) × x² × x × y² × y × z²

= - 72 × x³ × y³ × z²

= -72x³y³z²

× × × a × a² × b × b²

= × a⁶ × b³

= a⁶b³

× × × a × a × a² × b² × b × c² × c

= - a⁴ × b³ × c³

= -a⁴b³c³

× -5 × × u² × u × u × v × v × v² × w × w² × w

= × u⁴ × v⁴ × w⁴

= u⁴v⁴w⁴

0.5 × × 24 × x × x × y² × y × x² × z⁴ × z

= × x⁴ × y³ × z⁵

= 4x⁴ × y³ × z⁵

= 4x⁴y³z⁵

× × 16 × p × p² × p² × q² × q² × r × r²

= × p⁵ × q⁴ × r³

= p⁵q⁴r³

2.3 × 0.1 × o.16 × x × x × y

= 0.0368 × x² × y

= 0.0368x²y

3 × 4 × -5 × x × x × x

= -60 × x³

= -60x³

4 × -3 × × x² × x × x³

= × x⁶

= x⁶

5x⁴ × x⁶ × 4 × x²

= 5 × 4 × x⁴ × x⁶ × x²

= 20 × x¹²

= 20x¹²

x⁶ × 2x × (-4x) × 5

= 2 × -4 × 5 × x⁶ × x × x

= -40 × x⁸

= -40 x⁸

-8 × -2 × x² × x × y⁶ × y

= 16 × x³ × y⁷

= 16x³y⁷

Verification is when, x = 2.5 and y = 1

R.H.S = 16 (2.5)³ × (1)⁷

= 16 × 15.625

= 250

L.H.S = -8 × 2.5² × 1⁶ × -20 × 1 × 2.5

= 250

Therefore,

L.H.S = R.H.S

3.2 × 2.1 × x⁶ × x² × y³ × y²

= 6.72 × x⁸ × y⁵

= 6.72x⁸y⁵

Verify:

When x = 1 and y = 0.5

R.H.S = 6.72x³y⁵

= 6.72 × 1⁸ × 0.5⁵

= 0.21

L.H.S = 3.2 × 1⁶ × (-.5)³ × 2.1 × 1² × 0.5²
= 0.21

Therefore,

L.H.S = R.H.S

5 × -1.5 × -12 × x⁶ × x² × x × y³ × y²

= 90 × x⁹ × y⁵

= 90x⁹y⁵

Verification:

x = 1 and y = 0.5

R.H.S = 90x⁹y⁵

= 90 (1)⁹ (05)⁵

= 2.8125

L.H.S = 2.8125

Therefore,

L.H.S = R.H.S

2.3a⁵b² × 1.2a²b²

= 2.3 × 1.2 × a⁵ × a² × b² × b²

= 2.76 × a⁷ × b⁴

= 2.76a⁷b⁴

Verification:

a = 1 and b = 0.5

2.76 a⁷ b⁴ = 2.76 (1)⁷ (0.5)⁴

= 2.76 × 1 × 0.0025

= 0.1725

-8 × - 20 × x² × x × y⁶ × y

= 160x³y⁷

Verify:

When, x = 2.5 and y = 1

R.H.S = 160x³y⁷

= 160 × (2.5)³ × (1)⁷

= 2500

L.H.S = - 8 × 2.5² × 1 × -20 × 1 × 2.5

= 2500

Therefore,

L.H.S = R.H.S

-x × x³ × x × y³ × y × y

= -x⁵y⁵

Verify:

When x = 1 and y = 2

R.H.S = -x⁵y⁵

= (-1)⁵ × 2⁵

= -1 × 32

= -32

L.H.S = (-1) × 2³ × 2 × 1³ × 1 × 2

= - 32

Therefore,

L.H.S = R.H.S

× × 5 × x² × x⁴ × x × y⁴ × y² × y

= × x⁶ × y⁶

= x⁶y⁶

Verification:

When x = 1 and y = 2

R.H.S = × 1⁶ × 2⁶

= × 64

= 5 × 2

= 10

L.H.S = × 1² × 2⁴ × × 1⁴ × 2² × 1 × 2 × 5

= 10

Therefore,

L.H.S = R.H.S

× 15 × × a² × a × b × b² × c × c³

= 3 a³ × b³ × c³

= 3a³b³c³

× × × a² × a × b × b² × c × c²

= × a³ × b³ × c³

= a³b³c³

× × -8 × a × a³ × b × b² × b³ × c³ × c

= × a⁴ × b⁶ × c⁴

= a⁴b⁶c⁴

2 × × x × x² × x² × y × y² × y

= x⁵y⁵

= x⁵y⁵

Verification:

When x = 2 and y = 1

R.H.S = x⁵y⁵

= (2)⁵ × (-1)⁵

= × 32 × -1

= - 16

Therefore,

L.H.S = R.H.S

× × × x² × x × x² × y × y² × y²

= × x⁵ × y⁵

= x⁵y⁵

Verification:

When x = 2 and y = -1

R.H.S = x⁵y⁵

= (2)⁵ (-1)⁵

= × 32 × -1

= 56

Therefore,

L.H.S = R.H.S

2a³ (3a + 5b)

= 2a³ × 3a + 2a² × 5b

= 6 × a⁴ + 10a³b

-11a (3a + 2b)

= (-11a × 3a) + (-11a × 2b)

= -33a² – 2 × 11 × a × b

= -33a² – 22ab

-5a (7a – 2b)

= -5a × 7a – (-5a) × 2b

= -5 × 7 × a × a + 5 × 2 × a × b

= -35a² + 10ab

-11y² (3y + 7)

= -11y² × 3y – 11y² × 7

= -11 × 3 × y² × y – 11y² × 7

= -33y³ – 77y²

x (x³ + y³)

= x × x³ + x × y³

= x⁴ + xy³

xy (x³ – y³)

= xy × x³ – xy × y³

=x⁴y – xy⁴

0.1y (0.1x⁵ + 0.1y)

= 0.1y × 0.1x⁵ + 0.1y × 0.1y

= 0.01 × x⁵ × y + 0.01 × y²

= 0.01x⁵y + 0.01y²

(ab²c - a²c²) (-50a²b²c²)

= ab²c × -50a²b²c² - a²c² × -50a²b² × c²

= × 50 × a³ × b⁴ × c³ - × - 50 × a⁴ × b² × c⁴

= a³b⁴c³ + 12a⁴b²c⁴

= a³b⁴c³ + 12a⁴b²c⁴

xyz (xyz² - xy²z³)

= xyz × xyz² - xyz × xy²z³

= × x² × y² × z³ + × x² × y³ × z⁴

= x²y²z³ + x²y³z⁴

xyz (x²yz - xyz²)

= xyz × x²yz - xyz × xyz²

= × x³ × y² × z² + 9 × x² × y² × z³

= x³y²z² + 9x²y²z³

1.5x (10x²y – 100xy²)

= 1.5x × 10x²y – 1.5x × 100xy²

= 15 × x³ × y – 150 × x² × y²

= 15x³y – 150x²y²

4.1xy (1.1x – y)

= 4.1xy × 1.1x – 4.1xy × y

= 4.51x²y – 4.1xy²

250 × 5 (x²yz +

= 250 (5x²yz + )

= 250 × 5x²yz + 125xy²

(x³y³ + x³y)

= x³y³ + x³y

(a³ + ab² – 3ac²)

= a³ + ab² - ac²

24x² (1 – 2x)

= 24x² – 48x³

According to question,

When x = 3

= 24x² – 48x³

= 24 (3)² – 48 (3)³

= 24 (9) – 48 (27)

= 216 – 1296

= - 1080

- 3y (xy + y²)

= - 3xy² – 3y³

According to question:

When x = 4 and y = 5

= - 3xy² – 3y³

= - 3 (4) (5)² – 3 (5)³

= - 300 – 375

= - 675

= - 3x³y³bx + x²y⁴bx

= -3x⁴y³b + x³y⁴b

According to question:

When x = 1 and y = 2

= - 3 (1)⁴ (2)³ b + (1)³ (2)⁴ b

= - 3 (8) b + 3 (8) b

= 0

(i) 15y² (2 – 3x)

= 30y² – 45xy²

Putting x = -1, y = and z =

= 30 ()² – 45 (-1) ()²

= 30 () + 45 ()

= +

=

=

(ii) -3x (y²+z²)

Putting x = - 1, y = and z =

= - 3 (-1) ()² – 3 (-1) ()²

= +

= +

=

(iii) z² (x-y)

Putting x = - 1, y = and z =

z² (x – y)

= ()² (-1 - )

= () ()

=

(iv) xz(x+y²)

Putting x = - 1, y = and z =

= (-1)² () + (-1) ()² ()

= - ()

=

=

(5x + 3) × (7x + 2)

= 5x (7x + 2) + 3 (7x + 2)

= 35x² + 10x + 21x + 6

= 35x² + 31x + 6

(2x + 8) × (x – 3)

= 2x (x – 3) + 8 (x – 3)

= 2x² – 6x + 8x – 24

= 2x² – 2x - 24

(7x + y) × (x + 5y)

= 7x (x + 5y) + y (x + 5y)

= 7x² + 35xy + xy + 5y²

= 7x² + 36xy + 5y²

(a – 1) × (0.1a² + 3)

= a (0.1a² + 3) – 1 (0.1a² + 3)

= 0.1a³ + 3a – 0.1a² - 3

(3x² + y²) × (2x² + 3y²)

= 3x² (2x² + 3y²) + y² (2x² + 3y²)

= 6x⁴ + 9x²y² + 2x²y² + 3y⁴

= 6x⁴ + 11x²y² + 3y⁴

(x + y) × (x + 4y)

= x ( + 4y) + y (x + 4y)

= x² + xy + xy + 2y²

= ² + xy + 2y²

(x⁶ – y⁶) × (x² + y²)

= x⁶ (x² + y²) – y⁶ (x² + y²)

= x⁸ + x⁶y² – x²y⁶ – y⁸

(x² – y²) × (3a + 2b)

= x² (3a + 2b) – y² (3a + 2b)

= 3ax² + 2bx² – 3ay² – 2by²

(- 3d + 7f) × (5d + f)

= -3d (5d + f) + 7f (5d + f)

= - 15d² – 3df + 35df + 7f²

= - 15d² + 32df + 7f²

(0.8a – o.5b) × (1.5a – 3b)

= 0.8a (1.5a – 3b) – 0.5b (1.5a – 3b)

= 1.2a² – 2.4ab – 7.5ab + 1.5b²

= 1.2a² – 9.9ab + 1.5b

(2x²y² – 5xy²) × (x² – y²)

= 2x²y² (x² – y²) – 5xy² (x² – y²)

= 2x⁴y² – 2x²y⁴ – 5x³y² + 5xy⁴

( + ) × ( + )

= ( + ) + ( + )

= + + +

= + x² + x³

( + ) × ( – )

= ( – ) + ( – )

= + + –

(3x²y – 5xy²) × (x² + y²)

= 3x²y (x² + y²) – 5xy² (x² + y²)

= x⁴y + 3x²y³ – x³y² + xy⁴

(2x² – 1) × (4x³ + 5x²)

= 2x² (4x³ + 5x) – 1 (4x³ + 5x²)

= 8x⁵ + 10x³ – 4x³ – 5x²

= 8x⁵ + 6x³ – 5x²

(2xy + 3y²) × (3y² – 2)

= 2xy (3y² – 2) + 3y² (3y² – 2)

= 6xy³ – 4xy + 3y⁴ – 6y²

(3x – 5y) × (x + y)

= x (3x – 5y) + y (3x – 5y)

= 3x² – 5xy + 3xy – 5y²

= 3x² – 2xy – 5y²

Putting x = - 1 and y = - 2, we have

[3 (-1) – 5 (-2)] [(1) + (-2)] = 3 (-1)² – 2 (-1) (-2) – 5 (-2)²

(-3 + 10) (-1 – 2) = 3 – 4 – 20

- 21 = - 21

Therefore,

L.H.S = R.H.S

Hence, verified

x²y (3 – 2x²y) – 1 (3 – 2x²y)

= 3x²y- 2x⁴y² – 3 + 2x²y

= 2x⁴y² + 5x²y – 3

Putting x = -1 and y = -2, we have

= [(-1)² (-2) – 1] [3 – 2 (-1)² (-2) = [-2 (-1)⁴ (-2)² + 5 (-1)² (2) – 3]

= (-2 – 1) (3 + 4) = - 8 – 10 – 3

-21 = - 21

Therefore,

L.H.S = R.H.S

Hence, verified

(x)² – ()²

= (x – ) (x + )

= x² - y⁴

Putting x = -1 and y = -2, we have

((-1) – ) = ( (-1)² - )

= ( - ) ( + ) = ( - )

= () () =

= =

Therefore,

L.H.S = R.H.S

Hence, verified

x² (x² – 3xy + 2xy – 3y²)

= x² (x² – xy – 6y²)

= x⁴ – x³y – 6x²y²

(x³ + 4x²y – 2xy² – 8y³) × x²y²

= x⁵y² + 4x⁴y³ – 2x³y⁴ – 8x²y⁵

a²b² (3a² + ab + 6ab + 2b²)

= a²b² (3a² + 7ab + 2b²)

= 3a⁴b² + 7a³b³ + 2a²b⁴

x²y² (x – y) (x + 2y)

= x²y² (x² + 2xy – xy – 2y²)

= x²y² (x² + xy – 2y²)

= x⁴y² + x³y³ – 2x²y⁴

2x⁴ – 4x³ + 4x² – 14x – 3x³ + 6x² – 6x + 21

= 2x⁴ – 7x³ + 10x² – 20x + 21

(5x² – 2x – 3) (3x – 2)

= 15x³ – 6x² – 9x – 10x² + 4x + 6

= 15x³ – 16x² – 5x + 6

(x² – 7x + 10) (6 – 5x)

= -5x³ + 35x² – 50x + 6x² – 42x + 60

= -5x² + 41x² – 92x + 60

6x⁴ + 9x³ – 15x² – 10x³ – 15x² + 25x + 8x² + 12x – 20

= 6x⁴ – x³ – 22x² + 37x - 20

6x² – 9x – 4x + 6 + 5x² + 5x – 3x – 3

= 11x² – 11x + 3

5x² + 10x – 3x – 6 – 8x² + 6x – 20x + 15

= -3x² – 7x + 9

12x² + 9xy + 8xy

= 12x² + 9xy + 8xy + 6y² – 14x² + 6xy + 7xy – 3y²

= -2x² + 30xy + 3y²

5x⁴ – 15x² + 10x – 2x³ + 6x – 4 – (6x³ + 8x² – 10x – 3x² – 4x + 5)

= 5x⁴ – 15x² – 2x³ + 16x – 4 – 6x³ – 5x² + 14x – 5

= 5x⁴ – 8x³ – 20x² + 30x - 9

x⁴ – 2x³ + 3x² – 4x – x³ + 2x² – 3x + 4 – (2x³ – 2x² + 2x – 3x² + 3x – 3)

= x⁴ – 3x³ + 5x² – 7x + 4 – 2x³ + 5x² – 5x + 3

= x⁴ – 5x³ + 10x² – 12x + 7

(i)

x² + 2 (x) (2) + 2²

= x² + 4x + 4

(ii)

(8x)² + 2 (8x) (3b) + (3b)²

= 16x² + 48xb + 9b²

(iii)

(2m)² + 2 (2m) (1) + 1²

= 4m² + 4m + 1

(iv)

(9a)² + 2 (9a) () + (²

= 81a² + 3a +

(v)

(x)² + 2 (x) () + ()²

= x² + x³ + x⁴

(vi)

()² – 2 () () + ()²

= x² - + y²

(vii)

(3x)² – 2 (3x) () + ()²

= 9x² – 2 +

(viii)

()² – 2 () () + ()²

= - 2 +

(ix)

()² – 2 () () + ()²

= a² - ab + b

(x)

(a²b)² – 2 (a²b) (bc²) + (bc²)²

= a⁴b² – 2a²b²c² + b²c⁴

(xi)

()² + 2 () () + ()²

= + a +

(xii)

(x²)² – 2 (x²) (ay) + (ay)²

= x⁴ – 2x²ay + a²y²

(i) (2x + y) (2x + y)

2x (2x + y) + y (2x + y)

= 4x² + 2xy + 2xy + 3y

= 4x² + 4xy + 3y

(ii) (a + 2b) (a – 2b)

a (a – 2b) + 2b (a – 2b)

= a² – 2ab + 2ab – 4b²

= a² – 4b²

(iii)

a² (a² – bc) + bc (a² – bc)

= a⁴ – a²bc + bca² – b²c²

= a⁴ – b²c²

(iv)

( + ) - ( + )

= x² + yx - –

= x² - y²

(v)

2x (2x - ) + (2x - )

= 4x² - + -

= 4x² –

(vi)

2a³ (2a³ – b³) + b³ (2a³ – b³)

= 4a⁶ – 2a³b³ + 2a³b³ – b⁶

= 4a⁶ – b⁶

(vii)

x⁴ (x⁴ - ) + (x⁴ - )

= x⁸ – 2x² + 2x² -

= (x⁸ - )

(viii)

x³ (x³ - ) + (x³ - )

= x⁶ – 1 + 1 -

= x⁶ -

(i)

This can be written as:

(100 + 2)²

= (100)² + 2 (100) (2) + 2²

= 10000 + 400 + 4

= 10404

(ii)

This can be written as:

(100 – 1)²

= (100)² – 2 (100) (1) + 1²

= 10000 – 200 + 1

= 9801

(iii)

This can be written as:

(1000 + 1)²

= (1000)² + 2 (1000) (1) + 1²

= 1000000 + 2000 + 1

= 1002001

(iv)

This can be written as:

(1000 – 1)²

= (1000)² – 2 (1000) (1) + 1²

= 1000000 – 2000 + 1

= 998001

(v)

This can be written as:

(700 + 3)²

= (700)² + 2 (700) (3) + 3²

= 490000 + 4200 + 9

= 494209

(i)

Using formula:

(a – b) (a + b) = a² – b², we get

= (82 – 18) (82 + 18)

= 64 × 100

= 6400

(ii)

Using formula:

(a – b) (a + b) = a² – b², we get

= (467 – 33) (467 + 33)

= (434) (500)

= 217000

(iii)

Using formula:

(a – b) (a + b) = a² – b², we get

= (79 + 69) (79 – 69)

= (148) (10)

= 1480

(iv)

Using formula:

(a – b) (a + b) = a² – b², we get

= (200 – 3) (200 + 3)

= (200)² – (3)²

= 40000 – 9

= 39991

(v)

Using formula:

(a – b) (a + b) = a² – b², we get

= (100 + 3) (100 – 3)

= (100)² – (3)²

= 10000 – 9

= 9991

(vi)

Using formula:

(a – b) (a + b) = a² – b², we get

= (100 – 5) (100 + 5)

= (100)² – (5)²

= 10000 – 25

= 9975

(vii)

Using formula:

(a – b) (a + b) = a² – b², we get

= (2- 0.2) (2 + 0.2)

= (2)² – (0.2)²

= 4 – 0.04

= 3.96

(viii)

Using formula:

(a – b) (a + b) = a² – b², we get

= (10 – 0.2) (10 + 0.2)

= (10)² - (0.2)²

= 100 – 0.04

= 90.96

(i)

=

= 100

(ii)

(178)² – (22)²

= (178 + 22) (178 – 22)

= 200 × 156

= 31200

(iii)

=

= 300

(iv)

(1.73) – (0.27)

= (1.73 + 0.27) (1.73 – 0.27)

= 2 (1.46)

= 2.92

(v)

=

= 100

(i) 4x = 52² - 48²

4x = (52 – 48) (52 + 48)

4x = 4 × 100

4x = 400

x = 100

(ii)

14x = (47 – 33) (47 + 33)

14x = 14 × 80

x = 80

(iii)

Using formula:

a² – b² = (a – b) (a + b), we get

5x = (50 – 40) (50 + 40)

5x = 10 × 90

5x = 900

x = 180

Given that,

x + = 20

Squaring both sides, we get

(x + )² = (20)²

x² + 2 × x × + ()² = 400

x² + 2 + = 400

x² + = 398

(i) Given that,

x - = 3

Squaring both sides, we get

(x - )² = (3)²

x² - 2 × x × + ()² = 9

x² - 2 + = 9

x² + = 11

(ii) Squaring both sides, we get

(x² + )² = (11)²

(x²)² + 2 × x² × + ()² = 121

x⁴ + 2 + = 121

x⁴ + = 119

x² + = 18

Adding 2 on both sides, we get

x² + + 2 = 18 + 2

x² + + 2 × x × = 20

(x + )² = 20

x + = 2

Given that,

x² + = 18

Subtracting 2 from both sides, we get

x² + - 2 × x × = 18 – 2

(x - )² = 16

x - = 4

Given that,

x + y = 4 and xy=2

We take the equation: x + y = 4 and on squaring both sides, we get

(x + y)² = 4²

x² + y² + 2xy = 16

x² + y² + 2 (2) = 16 (Because xy=2 is given)

x² + y² + 4 = 16

x² + y² = 16 – 4

x² + y² =12

Therefore, the value of x² + y² is 12

Given that, x – y = 7

Squaring both sides, we get

(x – y)² = (7)²

x² + y² – 2xy = 49

Its given that xy = 9,

x² + y² – 2 (9) = 49

x² + y² = 49 + 18

x² + y² = 67

Given that,

3x + 5y = 11

Squaring both sides, we get

(3x + 5y)² = (11)²

(3x)² + (5y)² + 2 (3x) (5y) = 121

9x² + 25y² + 30xy = 121

9x² + 25y² + 30 (2) = 121

9x² + 25y² = 121 – 60

9x² + 25y² = 61

(i)

(4x)² + 2 (4x) (3) + 3²

= (4x + 3)²

Putting x =

= [4 () + 3]²

= (7 + 3)²

= 100

(ii)

(8x)² + 2 (8x) (9y) + (9y)²

= (8x + 9y)²

Putting x = 11 and y =

= [8 (11) + 9 ()]²

= (88 + 12)²

= (100)²

= 10000

(iii)

(9x)² + (4y)² – 2 (9x) (4y)

= (9x – 4y)²

Putting x = and y =

= [9 () – 4 ()]²

= (6 – 3)²

= 3²

= 9

Given that,

x + = 9

Squaring both sides, we get

(x + )² = 9²

x² + + 2 = 81

x² + = 79

Again,

Squaring both sides, we get

(x² + )² = 79²

x⁴ + + 2 = 6241

x⁴ + = 6239

Given that,

x + = 12

Squaring both sides, we get

(x + )² = 12²

x² + ()² + 2 × x × = 144

x² + = 142

Subtract 2 from both sides, we get

x² + - 2 × x × = 142 – 2

(x - )² = 140

x - = �

Given that,

2x + 3y = 14...............(1)

2x – 3y = 2..................(2)

Now, on squaring both the equation and subtracting (2) from (1), we get,

(2x + 3y)² – (2x – 3y)² = (14)² – (2)²

4x² + 9y² + 12xy – 4x² – 9y² + 12xy = 196 – 4

(The positive and negative terms gets cancelled)

24 xy = 192

xy = 8

Therefore, the value of "xy"is 8.

(i) x+y

Given that,

x²_{+ y}² = 29

x² + y² + 2xy – 2xy = 29

(x + y)² – 2 (2) = 29

(x + y)² = 29 + 4

x + y = �

(ii) x-y

x² + y² = 29

x² + y² + 2xy – 2xy = 29

(x – y)² + 2 (2) = 29

(x – y)² + 4 = 29

(x – y)² = 25

(x – y) = � 5

(iii) x⁴+y⁴

x² + y² = 29

Squaring both sides, we get

(x² + y²)² = (29)²

x⁴ + y⁴ + 2x²y² = 841

x⁴ + y⁴ + 2 (2)² = 841

x⁴ + y⁴ = 841 – 8

x⁴ + y⁴ = 833

(i)

_(2x)² – 2 (2x) (3) + 3² – 3² + 7

_{= (2x – 3)}² – 9 + 7

_{= (2x – 3)}² – 2

_{Hence, 2 must be added to the expression in order to make a whole square}

(ii)

(2x)² – 2 (2x) (5) + 5² – 5² + 20

= (2x – 5)² – 25 + 20

= (2x – 5)² – 5

Hence, 5 must be added to the expression in order to make it a whole square

(i)

(x² – y²) (x² + y²) (x⁴ + y⁴)

= [(x²)² – (y²)²] (x⁴ + y⁴)

= (x⁴ – y⁴) (x⁴ – y⁴)

= [(x⁴)² – (y⁴)²]

= x⁸ – y⁸

(ii)

[(2x)² – (1)²] (4x² + 1) (16x⁴ + 1)

= (4x² – 1) (4x² + 1) (16x⁴ + 1) 1

= [(4x²)² – (1)²] (16x⁴ + 1) 1

= (16x⁴ – 1) (16x⁴ + 1) 1

= [(16x⁴)² – (1)²] 1

= 256x⁸ – 1

(iii)

(7m)² + (8n)² – 112mn + (7m)² + (8n)² + 112mn

= 49m² + 64n² + 49m² + 64n²

= 98m² + 64n² + 64n²

= 98m² + 128n²

(iv)

(2.5p)² + (1.5q)² – 2 (2.5p) (1.5q) – (1.5p)² – (2.5q)² + 2 (1.5p) (2.5q)

= 6.25p² + 2.25q² – 2.25p² – 6.25q²

= 4p² – 6.25q² + 2.25q²

= 4p² – 4q²

= 4 (p² – q²)

(v)

_(m²)² – 2 (m²) (n²) (m) + (n²m)² + 2m³n²

_{= m}⁴ – 2m³n² + (n²m)² + 2m³n²

_{= m}⁴ + n⁴m² – 2m³n² + 2m³n²

_{= m}⁴ + m²n⁴

_{= m}² (m² + n⁴)

(i)

L.H.S = (3x + 7)² – 84x

= (3x)² + (7)² + 2 (3x) (7) – 84x

= (3x)² + (7)² + 42x – 84x

= (3x)² + (7)² – 42x

= (3x)² + (7)² – 2 (3x) (7)

= (3x – 7)²

= R.H.S

Hence, proved

(ii)

L.H.S = (9a – 5b)² + 180ab

= (9a)² + (5b)² – 2 (9a) (5b) + 180ab

= (9a)² 6 (5b)² – 90ab + 180ab

= (9a)² + (5b)² + 9ab

= (9a)² + (5b)² + 2 (9a) (5b)

= (9a + 5b)²

= R.H.S

Hence, proved

(iii)

L.H.S = ( - )² + 2mn

= ()² + ()² – 2mn + 2mn

= ()² + ()²

= m² + n²

= R.H.S

Hence, verified

(iv)

L.H.S = (4pq + 3q)² – (4pq – 3q)²

= (4pq)² + (3q)² + 2 (4pq) (3q) – (4pq)² – (3q)² + 24pq²

= 24pq² + 24pq²

= 48pq²

Hence, proved

(v)

L.H.S = (a – b) (a + b) + (b – c) (b + c) + (c – a) (c + a)

Using identity:

(a – b) (a + b) = a² – b²

We get,

= (a² – b²) + (b² – c²) + (c² – a²)

= a² – b² + b² – c² + c² – a²

= 0

= R.H.S

Hence, verified

(i)

x (x + 7) + 4 (x + 7)

= x² + 7x + 4x + 28

= x² + 11x + 28

(ii)

x (x + 4) – 11 (x + 4)

= x² + 4x – 11x – 44

= x² – 7x - 44

(iii)

x (x – 5) + 7 (x – 5)

= x² – 5x + 7x – 35

= x² + 2x - 35

(iv)

x (x – 2) – 3 (x – 2)

= x² – 2x – 3x + 6

= x² – 5x + 6

(v)

y² (y² – 3) – 4 (y² – 3)

= y⁴ – 3y² – 4y² + 12

= y⁴ – 7y² + 12

(vi)

x (x + ) + (x + )

= x² + + +

= x² + + + 1

= x² + + 1

(vii)

3x (3x + 11) + 5 (3x + 11)

= 9x² + 33x + 15x + 55

= 9x² + 48x + 55

(viii)

2x² (2x² – 5) – 3 (2x² – 5)

= 4x⁴ – 10x² – 6x² + 15

= 4x⁴ – 16x² + 15

(ix)

z² (z² – 3) + 2 (z² – 3)

= z⁴ – 3z² + 2z² – 6

= z⁴ – z² - 6

(x)

3x (2x – 4y) – 4y (2x – 4y)

= 6x² – 12xy – 8xy + 16y²

= 6x² – 20xy + 16y²

(xi)

3x² (3x² – 3xy) – 4xy (3x² – 3xy)

= 9x⁴ – 9x³y – 12x³y + 12x²y²

= 9x⁴ – 21x³y + 12x²y²

(xii)

x (x + ) + 5 (x + )

= x² + + 5x + 1

= x² + x + 1

(xiii)

z (z + ) + (z + )

= z² + z + z +

= z² + z + z + 1

= z² + z + 1

(xiv)

x² (x² + 9) + 4 (x² + 9)

= x⁴ + 9x² + 4x² + 36

= x⁴ + 13x² + 36

(xv)

y² (y² + 6) + 12 (y² + 6)

= y⁴ + 6y² + 12y² + 72

= y⁴ + 18y² + 72

(xvi) (y² + ) (y² - )

y² (y² - ) + (y² - )

= y⁴ - y² + y² – 2

= y⁴ - y² - 2

(xvii)

p² (p² - ) + 16 (p² - )

= p⁴ – p² + 16p² – 4

= p⁴ - p² - 4

(i)

(100 + 2) (100 + 6)

= 100 (100 + 6) + 2 (100 + 6)

= 10000 + 600 + 200 + 12

= 10812

(ii)

This can be written as:

(100 + 9) (100 + 7)

= 100 (100 + 7) + 9 (100 + 7)

= 10000 + 700 + 900 + 63

= 11663

(iii)

This can be written as:

(30 + 5) (30 + 7)

= 30 (30 + 7) + 5 (30 + 7)

= 900 + 210 + 150 + 35

= 1295

(iv)

This can be written as:

(50 + 3) (50 + 5)

= 50 (50 + 5) + 3 (50 + 5)

= 2500 + 250 + 150 + 15

= 2915

(v)

This can be written as:

(100 + 3) (100 - 4)

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

= 10000 - 400 + 300 - 12

= 10000 – 112

= 9888

(vi)

This can be written as:

(30 + 4) (30 + 6)

= 30 (30 + 6) + 4 (30 + 6)

= 900 + 180 + 120 + 24

= 1224

(vii)

This can be written as:

(1000 - 6) (1000 + 6)

= 1000 (1000 + 6) - 6 (1000 + 6)

= 1000000 + 6000 - 6000 - 36

= 999964