R d sharma solutions mathematics class 8 Solutions for Chapter 6 algebraic expressions and identities

Identify the terms, their coefficients for each of the following expressions:

(i) 7x2yz - 5xy

(ii) x2 + x + 1

(iii) 3x2y2– 5x2y2z2+ z2

(iv) 9 – ab + bc - ca

(v) a/2 + b/2 - ab

(vi) 0.2x – 0.3xy + 0.5y

Classify the following polynomials as monomials, binomials, trinomials. Which polynomials do not fit in any category?

(i) x+y

(ii) 1000

(iii) x+x2+x3+x4

(iv) 7+a+5b

(v) 2b-3b2

(vi) 2y-3y2+4y3

(vii) 5x-4y+3x

(viii) 4a-15a2

(ix) xy+yz+zt+tx

(x) pqr

(xi) p2q+pq2

(xii) 2p+2q

Add the following algebraic expressions:

(i) 3a2 b, -4a2 b, 9a2 b

(ii) 2/3a, 3/5a, -6/5a

(iii) 4xy2 – 7x2 y, 12x2 y -6xy2 , -3x2 y + 5xy2

(iv) 3/2a – 5/4b + 2/5c, 2/3a - 7/2b + 7/2c, 5/3a + 5/2b – 5/4c

(v) 11/2xy + 12/5y + 13/7x, -11/2y – 12/5x – 137xy

(vi) 7/2x3 – 1/2x2 + 5/3, 3/2x3 + 7/4x2 – x + 1/3, 3/2x2 -5/2x -2

Subtract:

(i) -5xy from 12xy

(ii) 2a2 from -7a2

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

(iv) 2x3 - 4x2 + 3x + 5 from 4x3 + x2 + x + 6

(v) 3/2y3 – 2/7y2– 5 from 1/3y3+ 5/7y2 + y – 2

(vi) 3/2x – 5/4y – 7/2z from 2/3x + 3/2y – 4/3z

(vii) x2 y – 4/5xy2 + 4/3xy from 2/3x2 y + 3/2xy2 -1/3xy

(viii) ab/7 -35/3bc + 6/5ac from 3/5bc – 4/5ac

Take away:

(i) 6/5x2 -4/5x3 + 5/6 + 3/2x from x3 /3 – 5/2x2 + 3/5x + 1/4

(ii) 5a2 /2 + 3a3 /2 + a/3 – 6/5 from 1/3a3 – 3/4a2 – 5/2

(iii) 7/4x3 + 3/5x2 + 1/2x + 9/2 from 7/2 – x/3 – x 2 /5

(iv) y3 /3 + 7/3y2 + 1/2y + 1/2 from 1/3 – 5/3y2

(v) 2/3ac – 5/7ab + 2/3bc from 3/2ab -7/4ac – 5/6bc

Subtract 3x - 4y - 7z from the sum of x - 3y + 2z and -4x + 9y - 11z.

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

Subtract the sum of 2x - x 2 + 5 and -4x – 3 + 7x2 from 5.

Simplify each of the following: (i) x2 – 3x + 5 – 1/2(3x2 -5x + 7)

(ii) [[5 – 3x + 2y – (2x - y)]] – (3x – 7y + 9)

(iii) 11/2x2 y – 9/4xy2 + 1/4xy – 1/14y2 x + 1/15yx2 + 1/2xy (iv) (1/3y2 – 4/7y + 11) – (1/7y – 3 + 2y2 ) – (2/7y – 2/3y2 + 2)

(v) -1/2a2 b2 c + 1/3ab2 c – 1/4abc2 – 1/5cb2 a2 + 1/6cb2 a – 1/7c2 ab + 1/8ca2 b

Find each of the following products:

5x2 × 4x3

Find each of the following products:

-3a2 × 4b4

Find each of the following products:

(-5xy) × (-3x2 yz)

Find each of the following products:

1/2xy × 2/3x2 yz 2

Find each of the following products:

(-7/5xy2 z) × (13/3x2 yz 2 )

Find each of the following products:

(-24/25x3 z) × (-15/16xz2 y)

Find each of the following products:

(-1/27a2 b2 ) × (9/2a3 b2 c 2 )

Find each of the following products:

(-7xy) × (1/4x2 yz)

Find each of the following products:

(7ab) × (-5ab2 c) × (6abc2 )

Find each of the following products:

(-5a) × (-10a2 ) × (-2a3 )

Find each of the following products:

(-4x2 ) × (-6xy2 ) × (-3yz2 )

Find each of the following products:

(7/9ab2 ) × (15/7ac2 b) × (-3/5a2 c)

Find each of the following products:

(4/3u2 vw) × (-5uvw2 ) × (1/3v2 wu)

Find each of the following products:

(0.5x) × (1/3xy2 z 4 ) × (24x2 yz)

Find each of the following products:

(4/3pq2 ) × (-1/4p2 r) × (16p2 q2 r 2 )

Find each of the following products:

(2.3xy) × (0.1x) × (0.16)

Express each of the following products as a monomials and verify the result in each case for x=1:

(3x) × (4x) × (-5x)

Express each of the following products as a monomials and verify the result in each case for x=1:

(4x2 ) × (-3x) × (4/5x3 )

Express each of the following products as a monomials and verify the result in each case for x=1:

(5x4 ) × (x2 ) 3 × (2x) 2

Express each of the following products as a monomials and verify the result in each case for x=1:

(x2 ) 3 × (2x) × (-4x) × (5)

Write down the product of -8x2 y 6 and -20xy verify the product for x = 2.5, y = 1

Evaluate (3.2x6 y 3 ) × (2.1x2 y 2 ) when x = 1 and y = 0.5

Find the value of (5x6 ) × (-1.5x2 y 3 ) × (-12xy2 ) when x = 1, y = 0.5

Evaluate (2.3a5 b2 ) × (1.2a2 b2 ) when a = 1 and b = 0.5

Evaluate (-8x2 y 6 ) × (-20xy) for x = 2.5 and y = 1

Express each of the following products as a monomials and verify the result for x = 1, y = 2:

(-xy3 ) × (yx3 ) × (xy)

Express each of the following products as a monomials and verify the result for x = 1, y = 2:

(1/8x2 y 4 ) × (1/4x4 y 2 ) × (xy) × 5

Express each of the following products as a monomials and verify the result for x = 1, y = 2:

(2/5a2 b) × (-15b2 ac) × (-1/2c2 )

Express each of the following products as a monomials and verify the result for x = 1, y = 2:

(-4/7a2 b) × (-2/3b2 c) × (-7/6c2 a)

Express each of the following products as a monomials and verify the result for x = 1, y = 2:

(4/9abc3 ) × (-27/5a3 b2 ) × (-8b3 c)

Evaluate each of the following when x = 2, y = -1.

(2xy) × (x2 y/4) × (x2 ) × (y2 )

Evaluate each of the following when x = 2, y = -1.

(3/5x2 y) × (-15/4xy2 ) × (7/9x2 y 2 )

Find the following products:

2a 3 (3a + 5b)

Find the following products:

-11a (3a + 2b)

Find the following products:

-5a (7a - 2b)

Find the following products:

-11y 2 (3y + 7)

Find the following products:

6x/5(x 3 + y 3 )

Find the following products:

xy (x3 -y 3 )

Find the following products:

0.1y (0.1x5 + 0.1y)

Find the following products:

(-7/4ab2 c – 6/25a2 c 2 ) (-50a2 b2 c 2 )

Find the following products:

-8/27xyz (3/2xyz2 – 9/4xy2 z 3 )

Find the following products:

-4/27xyz (9/2x2 yz – 3/4xyz2 )

Find the following products:

1.5x (10x2 y – 100xy2 )

Find each of the following products:

(-2/7a4 ) × (-3/4a2 b) × (-14/5b2 )

Find the following products:

4.1xy (1.1x – y)

Find the following products:

250.5xy (xz + y/10)

Find the following products:

7/5x2 y (3/5xy2 + 2/5x)

Find the following products:

4/3a (a2 + b2 – 3c2 )

Find the product 24x2 (1-2x) and evaluate its value for x = 3

Find the product -3y (xy+y2 ) and evaluate its value for x = 4 and y = 5

Multiply -3/2x2 y 3 by (2x-y) and verify the answer for x = 1 and y = 2

Multiply the monomial by the binomial and find the value of each for x = -1, y = 0.25 and z = 0.005:

(i) 15y2 (2 - 3x)

(ii) -3x (y2 +z2 )

(iii) z 2 (x - y)

(iv) xz (x 2 + y 2 )

Simplify:

(i) 2x 2 (x3 -x) – 3x (x4 + 2x) – 2 (x4 – 3x2 )

(ii) x3 y (x2 -2x) + 2xy (x3 – x 4 )

(iii) 3a2 + 2 (a+2) – 3a (2a+1)

(iv) x (x+4) + 3x (2x2 -1) + 4x2 + 4

(v) a (b-c) – b (c-a) – c (a-b)

(vi) a (b-c) +b (c-a) + c (a-b)

(vii) 4ab (a-b) – 6a2 (b-b2 ) -3b2 (2a2 -a) + 2ab (b-a)

(viii) x2 (x2 + 1) – x 3 (x + 1) – x (x3 -x)

(ix) 2a2 + 3a (1 – 2a3 ) + a (a + 1)

(x) a2 (2a - 1) + 3a + a3 -8

(xi) 3/2x2 (x2

-1) + 1/4x2 (x2 + x) - 3/4x (x3 -1)

(xii) a2 b (a-b2 ) + ab2 (4ab – 2a2 ) – a3 b(1-2b)

(xiii) a2 b (a3 -a + 1) - ab(a4 – 2a2 + 2a) – b(a3 -a2 -1)

Multiply:

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

Multipy:

(a - 1) by (0.1a2 + 3)

Multiply:

(2x + 8) by (x - 3)

Multiply:

(7x + y) by (x + 5y)

Multipy:

(3x2 + y2 ) by (2x2 + 3y2 )

Multipy:

(3/5x + 1/2y) by (5/6x + 4y)

Multiply:

(x6 – y 6 ) by (x2 + y2 )

Multipy:

(x2 + y2 ) by (3a + 2b)

Multipy:

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

Multipy:

(0.8a – o.5b) by (1.5a – 3b)

Multipy:'

(2x2 y 2 – 5xy2 ) by (x2 – y 2 )

Mutlipy:

(x/7 + x2 /2) by (2/5 + 9x/4)

Multipy:

(-a/7 + a2 /9) by (b/2 – b2 /3)

Mutlipy:

(3x2 y – 5xy2 ) by (1/5x2 + 1/3y2 )

Multipy:

(2x2 – 1) by (4x3 + 5x2 )

Multipy:

(2xy + 3y2 ) by (3y2 – 2)

Find the following products and verify the results for x = -1, y = -2:

(3x – 5y) (x + y)

Find the following products and verify the results for x = -1, y = -2:

(x2 y - 1) (3 – 2x2 y)

Find the following products and verify the results for x = -1, y = -2:

(1/3x – y 2 /5) (1/3x + y2 /5)

Simplify:

x 2 (x + 2y) (x – 3y)

Simplify:

(x

2

-2y2

) (x + 4y)x2

y

2

Simplify:

a2 b2 (a + 2b) (3a + b)

Simplify:

x 2 (x - y) y2 (x + 2y)

Simplify:

(x 3 – 2x2 + 5x - 7) (2x - 3)

Simplify:

(5x + 3) (x - 1) (3x - 2)

Simplify:

(5 - x) (6 – 5x) (2 - x)

Simplify:

(2x2 + 3x - 5) (3x2 -5x + 4)

Simplify:

(3x - 2) (2x - 3) + (5x - 3) (x + 1)

Simplify:

(5x - 3) (x + 2) - (2x + 5) (4x - 3)

Simplify:

(3x + 2y) (4x + 3y) - (2x - y) (7x – 3y)

Simplify:

(x2 -3x + 2) (5x - 2) - (3x2 + 4x - 5) (2x – 1)

Simplify:

(x 3 -2x2 + 3x - 4) (x - 1) - (2x – 3) (x2 -x + 1)

Write the following squares of binomials as trinomials:

(i) (x + 2)2

(ii) (8a + 3b)2

(iii) (2m + 1)2

(iv) (9a + 1/6)2

(v) (x + x 2 /2) 2

(vi) (x/4 – y/3)2

(vii) (3x – 1/3x)2

(viii) (x/y – y/x)2

(ix) (3a/2 – 5b/4)2

(x) (a2b – bc2)2

(xi) (2a/3b + 2b/3a)2

(xii) (x 2 – ay)2

Find the product of the following binomials:

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

(ii) (a + 2b) (a – 2b)

(iii) (a2 + bc) (a2 – bc)

(iv) (4x/5 – 3y/4) (4x/5 + 3y/4)

(v) (2x + 3/y) (2x - 3/y)

(vi) (2a3 + b3 ) (2a3 – b3 )

(vii) (x4 + 2/x2 ) (x4 -2/x2 )

(viii) (x3 + 1/x3 ) (x3 -1/x3 )

Using the formula for squaring a binomial, evaluate the following:

(i) (102)2

(ii) (99)2

(iii) (1001)2

(iv) (999)2

(v) (703)2

Simplify the following using the formula: (a - b) (a + b) = a2 – b2 :

(i) (82)2 – (18)2

(ii) (467)2 – (33)2

(iii) (79)2 – (69)2

(iv) 197 × 203

(v) 113 × 87

(vi) 95 × 105

(vii) 1.8 × 2.2

(viii) 9.8 × 10.2

Simplify the following using the identities:

(i) ((58)2 – (42)2 )/16

(ii) 178 × 178 – 22 × 22

(iii) (198 × 198 – 102 × 102)/96

(iv) 1.73 × 1.73 – 0.27 × 0.27

(v) (8.63 × 8.63 – 1.37 × 1.37)/0.726

Find the value of x, if:

(i) 4x = (52)2 – (48)2

(ii) 14x = (47)2 – (33)2

(iii) 5x = (50)2 – (40)2

If x + 1/x =20, find the value of x2 + 1/ x 2 .

If x – 1/x = 3, find the values of x2 + 1/ x2 and x4 + 1/ x 4

If x2 + 1/x2 = 18, find the values of x + 1/ x and x - 1/ x.

If x + y = 4 and xy = 2, find the value of x2 +y2

If x - y = 7 and xy = 9, find the value of x2 +y2

If 3x + 5y = 11 and xy = 2, find the value of 9x2 +25y2

Find the values of the following expressions:

(i) 16x2 + 24x + 9 when x = 7/4

(ii) 64x2 + 81y2 + 144xy when x = 11 and y = 4/3

(iii) 81x2 + 16y2 -72xy when x = 2/3 and y = ¾

If x + 1/x = 9 find the value of x4 + 1/ x 4

If x + 1/x = 12 find the value of x - 1/x.

If 2x + 3y = 14 and 2x - 3y = 2, find value of xy. [Hint: Use (2x+3y) 2 – (2x-3y) 2 = 24xy]

If x2 + y2 = 29 and xy = 2, find the value of

(i) x + y

(ii) x - y

(iii) x 4 +y4

What must be added each of the following expression to make it a whole square?

(i) 4x 2 – 12x + 7

(ii) 4x 2 – 20x + 20

Simplify:

(i) (x - y) (x + y) (x2 + y2 ) (x4 + y4 )

(ii) (2x - 1) (2x + 1) (4x2 + 1) (16x4 + 1)

(iii) (7m – 8n) 2 + (7m + 8n)2

(iv) (2.5p – 1.5q) 2 -(1.5p – 2.5q) 2

(v) (m2 –n2 m) 2 + 2m3 n2

Show that:

(i) (3x + 7)2 – 84x = (3x - 7)2

(ii) (9a – 5b)2 + 180ab = (9a + 5b)2

(iii) (4m/3 – 3n/4)2 + 2mn = 16m2 /9 + 9n2 /16

(iv) (4pq + 3q)2 – (4pq - 3q)2 = 48pq2 (v) (a - b) (a + b) + (b - c) (b + c) + (c - a) (c + a) = 0

Find the following products:

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

(ii) (x - 11) (x + 4)

(iii) (x + 7) (x - 5)

(iv) (x - 3) (x - 2 ) (v) (y 2 -4) (y 2 -3)

(vi)(x + 4/3) (x + 3/4)

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

(viii) (2 x 2 -3) (2x 2 + 5 )

(ix) ( z 2 + 2) (z2 -3)

(x) (3x – 4y) (2x – 4y)

(xi) (3 x 2 – 4xy) (3x 2 – 3xy)

(xii) (x + 1/5) (x + 5)

(xiii) (z + 3/4) (z + 4/3)

(xiv) (x 2 + 4) (x 2 + 9 )

(xv) (y 2 + 12) (y 2 + 6)

(xvi) (y 2 + 5/7) (y 2 – 14/5) (xvii) (p 2 + 16) (p 2 – 1/4)

Evaluate the following:

(i) 102 × 106

(ii) 109 × 107

(iii) 35 × 37

(iv) 53 × 55

(v) 103 × 96

(vi) 34 × 36

(vii) 994 × 1006