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**Add the following: (i) 3 x and 7x (ii) âˆ’5xy and 9xy**

We have

(i) 3x + 7x = (3 + 7)x = 10x

(ii) -5xy + 9xy = ( -5 + 9)xy = 4xy

**Simplify each of the following: (i) 7 x^{3}y + 9yx^{3} (ii) 12a^{2}b + 3ba^{2}**

Simplifying the given expressions, we have

(i) 7x^{3}y + 9yx^{3} = (7 + 9)x^{3}y = 16x^{3}y

(ii) 12a^{2}b + 3ba^{2} = (12 + 3)a^{2}b = 15a^{2}b

**Add the following: (i) 7 abc, âˆ’5abc, 9abc, âˆ’8abc (ii) 2x^{2}y, âˆ’ 4x^{2}y, 6x^{2}y, âˆ’5x^{2}y**

Adding the given terms, we have

(i) 7abc + (- 5abc) + (9 abc) + (- 8abc)

= 7abc - 5abc + 9abc - 8abc

= (7 - 5 + 9 - 8)abc

= (16 - 13)abc

= 3abc

(ii) 2x^{2}y + (- 4x^{2}y) + 6x^{2}y + (- 5x^{2}y)

= 2x^{2}y - 4x^{2}y + 6x^{2}y - 5x^{2}y

= (2 - 4 + 6 - 5)x^{2}y

= (8 - 9)x^{2}y

= -x^{2}y

**Add the following expressions: (i) x ^{3}âˆ’2x^{2}y+3xy^{2}âˆ’y^{3}, 2x^{3}âˆ’5xy^{2}+3x^{2}yâˆ’4y^{3}x^{3}-2x^{2}y+3xy^{2}-y^{3}, 2x^{3}-5xy^{2}+3x^{2}y-4y^{3} (ii) a^{4}âˆ’2a^{3}b+3ab^{3}+4a^{2}b^{2}+3b^{4},âˆ’2a^{4}âˆ’5ab^{3}+7a^{3}bâˆ’6a^{2}b^{2}+b^{4}a^{4}-2a^{3}b+3ab^{3}+4a^{2}b^{2}+3b^{4},-2a^{4}-5ab^{3}+7a^{3}b-6a^{2}b^{2}+b^{4}**

Adding the given expressions, we have

(i) x^{3}- 2x^{2}y + 3xy^{2}- y^{3}+ 2x^{3}- 5xy^{2} + 3x^{2}y- 4y^{3}

Collecting positive and negative like terms together, we get

x^{3}+ 2x^{3}- 2x^{2}y + 3x^{2}y + 3xy^{2}- 5xy^{2}^{ }- y^{3} - 4y^{3}

= 3x^{3} + x^{2}y - 2xy^{2} - 5y^{3}

(ii) (a^{4}- 2a^{3}b + 3ab^{3} + 4a^{2}b^{2} + 3b^{4}) + (-2a^{4}- 5ab^{3} + 7a^{3}b - 6a^{2}b^{2} + b^{4})

a^{4}- 2a^{3}b + 3ab^{3} + 4a^{2}b^{2} + 3b^{4} - 2a^{4}- 5ab^{3} + 7a^{3}b - 6a^{2}b^{2} + b^{4}

Collecting positive and negative like terms together, we get

a^{4} - 2a^{4} - 2a^{3}b + 7a^{3}b + 3ab^{3} - 5ab^{3} + 4a^{2}b^{2} - 6a^{2}b^{2} + 3b^{4} + b^{4}

= - a^{4} + 5a^{3}b - 2ab^{3} - 2a^{2}b^{2} + 4b^{4}

**Add the following expressions: (i) 8aâˆ’6ab+5b, âˆ’6aâˆ’abâˆ’8b and âˆ’4a+2ab+3b8a-6ab+5b, -6a-ab-8b and -4a+2ab+3b (ii) 5x3+7+6xâˆ’5x2, 2x2âˆ’8âˆ’9x, 4xâˆ’2x2+3x3, 3x3âˆ’9xâˆ’x2 and xâˆ’x2âˆ’x3âˆ’45x3+7+6x-5x2, 2x2-8-9x, 4x-2x2+3x3, 3x3-9x-x2 and x-x2-x3-4**

(i) Required expression = (8a - 6ab + 5b) + (- 6a - ab - 8b) + ( - 4a + 2ab + 3b)

Collecting positive and negative like terms together, we get

8a - 6a - 4a - 6ab - ab + 2ab + 5b - 8b + 3b

= 8a - 10a - 7ab + 2ab + 8b - 8b

= - 2a - 5ab

(ii) Required expression = (5x^{3} + 7 + 6x - 5x^{2}) + (2x^{2}^{ }- 8 - 9x) + (4x - 2x^{2} + 3x^{3}) + (3x^{3}- 9x - x^{2}) + ( x - x^{2}- x^{3}- 4)

Collecting positive and negative like terms together, we get

5x^{3}+ 3x^{3} + 3x^{3}- x^{3}- 5x^{2} + 2x^{2} - 2x^{2} - x^{2}- x^{2} + 6x - 9x + 4x - 9x + x + 7 - 8 - 4

= 11x^{3} - x^{3} - 7x^{2} + 11x - 18x + 7 - 12

= 10x^{3} - 7x^{2} - 7x - 5

**Add the following: (i)xâˆ’3yâˆ’2z 5x+7yâˆ’8z 3xâˆ’2y+5z (ii)4abâˆ’5bc+7ca âˆ’3ab+2bcâˆ’3ca 5abâˆ’3bc+4ca**

(i) Required expression = (x - 3y - 2z) + (5x +7y - 8z) +(3x - 2y + 5z)

Collecting positive and negative like terms together, we get

x + 5x + 3x - 3y + 7y - 2y - 2z - 8z + 5z

= 9x - 5y + 7y - 10z + 5z

= 9x + 2y - 5z

(ii) Required expression = (4ab - 5bc + 7ca) + (- 3ab + 2bc - 3ca ) + (5ab - 3bc + 4ca)

Collecting positive and negative like terms together, we get

4ab - 3ab + 5ab - 5bc + 2bc - 3bc + 7ca - 3ca + 4ca

= 9ab - 3ab - 8bc + 2bc + 11 ca - 3ca

= 6ab - 6bc + 8ca

**Add 2 x^{2} âˆ’ 3x + 1 to the sum of 3x^{2} âˆ’ 2x and 3x + 7.**

Sum of 3x^{2} - 2x and 3x + 7

= (3x^{2} - 2x) + ( 3x +7)

= 3x^{2} - 2x + 3x + 7

= (3x^{2} + x + 7)

Now, required expression = (2x^{2} - 3x + 1) + (3x^{2} + x + 7)

= 2x^{2} + 3x^{2 }- 3x + x + 1 + 7

= 5x^{2} - 2x + 8

**Add x^{2} + 2xy + y^{2} to the sum of x^{2} âˆ’ 3y^{2} and 2x^{2} âˆ’ y^{2}+ 9.**

Sum of x^{2} - 3y^{2} and 2x^{2} - y^{2} + 9

= (x^{2} - 3y^{2}) + (2x^{2} - y^{2} + 9)

= x^{2} + 2x^{2} - 3y^{2} - y^{2}+ 9

= 3x^{2} - 4y^{2} + 9

Now, required expression = (x^{2} + 2xy + y^{2}) + (3x^{2} - 4y^{2} + 9)

= x^{2} + 3x^{2} + 2xy + y^{2} - 4y^{2} + 9

= 4x^{2} + 2xy - 3y^{2}^{ }+ 9