R.D. Sharma solutions for Mathematics [English] Class 8 chapter 6 - Algebraic Expressions and Identities [Latest edition]

Identify the term, their coefficients for the following expression:

 7x²yz − 5xy

Identify the term, their coefficients for the following expression:

 x² + x + 1

Identify the term, their coefficients for the following expression:

 3x²y² − 5x²y²z² + z²

Identify the term, their coefficients for the following expression:

9 − ab + bc − ca

Identify the term, their coefficients for the following expression:

\[\frac{a}{2} + \frac{b}{2} - ab\]

Identify the term, their coefficients for the following expression:

0.2x − 0.3xy + 0.5y

Classify the following polynomial as monomial, binomial, trinomial. Which polynomial do not fit in any category?

x + y

Classify the following polynomial as monomial, binomial, trinomial. Which polynomial do not fit in any category?

 1000

Classify the following polynomial as monomial, binomial, trinomial. Which polynomial do not fit in any category?

x + x² + x³ + 4y⁴

Classify the following polynomial as monomial, binomial, trinomial. Which polynomial do not fit in any category?

 7 + a + 5b

Classify the following polynomial as monomial, binomial, trinomial. Which polynomial do not fit in any category?

2b − 3b²

Classify the following polynomial as monomial, binomial, trinomial. Which polynomial do not fit in any category?

 2y − 3y² + 4y³

Classify the following polynomial as monomial, binomial, trinomial. Which polynomial do not fit in any category?

5x − 4y + 3x

Classify the following polynomial as monomial, binomial, trinomial. Which polynomial do not fit in any category?

 4a − 15a²

Classify the following polynomial as monomial, binomial, trinomial. Which polynomial do not fit in any category?

xy + yz + zt + tx

Classify the following polynomial as monomial, binomial, trinomial. Which polynomial do not fit in any category?

 pqr

Classify the following polynomial as monomial, binomial, trinomial. Which polynomial do not fit in any category?

p²q + pq²

Classify the following polynomial as monomial, binomial, trinomial. Which polynomial do not fit in any category?

2p + 2q

Add the following algebraic expression:

3a²b, − 4a²b, 9a²b

Add the following algebraic expression:

\[\frac{2}{3}a, \frac{3}{5}a, - \frac{6}{5}a\]

Add the following algebraic expression:

 4xy² − 7x²y, 12x²y − 6xy², − 3x²y +5xy²

Add the following algebraic expression:

\[\frac{3}{2}a - \frac{5}{4}b + \frac{2}{5}c, \frac{2}{3}a - \frac{7}{2}b + \frac{7}{2}c, \frac{5}{3}a + \frac{5}{2}b - \frac{5}{4}c\]

Add the following algebraic expression:

\[\frac{11}{2}xy + \frac{12}{5}y + \frac{13}{7}x, - \frac{11}{2}y - \frac{12}{5}x - \frac{13}{7}xy\]

Add the following algebraic expression: \[\frac{7}{2} x^3 - \frac{1}{2} x^2 + \frac{5}{3}, \frac{3}{2} x^3 + \frac{7}{4} x^2 - x + \frac{1}{3}, \frac{3}{2} x^2 - \frac{5}{2}x - 2\]

Subtract:

− 5xy from 12xy

Subtract:

2a² from − 7a²

Subtract:

2a − b from 3a − 5b

Subtract:

2x³ − 4x² + 3x + 5 from 4x³ + x² + x + 6

Subtract:

\[\frac{2}{3} y^3 - \frac{2}{7} y^2 - 5 \text { from  }\frac{1}{3} y^3 + \frac{5}{7} y^2 + y - 2\]

Subtract: 

\[\frac{3}{2}x - \frac{5}{4}y - \frac{7}{2}z \text { from }\frac{2}{3}x + \frac{3}{2}y - \frac{4}{3}z\]

Subtract: 

\[x^2 y - \frac{4}{5}x y^2 + \frac{4}{3}xy \text { from } \frac{2}{3} x^2 y + \frac{3}{2}x y^2 - \frac{1}{3}xy\]

Subtract: 

\[\frac{ab}{7} - \frac{35}{3}bc + \frac{6}{5}ac \text { from } \frac{3}{5}bc - \frac{4}{5}ac\]

Take away:

\[\frac{6}{5} x^2 - \frac{4}{5} x^3 + \frac{5}{6} + \frac{3}{2}x \text { from  }\frac{x^3}{3} - \frac{5}{2} x^2 + \frac{3}{5}x + \frac{1}{4}\]

Take away:

\[\frac{5 a^2}{2} + \frac{3 a^3}{2} + \frac{a}{3} - \frac{6}{5} \text { from } \frac{1}{3} a^3 - \frac{3}{4} a^2 - \frac{5}{2}\]

Take away: 

\[\frac{7}{4} x^3 + \frac{3}{5} x^2 + \frac{1}{2}x + \frac{9}{2}\text {  from } \frac{7}{2} - \frac{x}{3} - \frac{x^2}{5}\]

Take away: 

\[\frac{y^3}{3} + \frac{7}{3} y^2 + \frac{1}{2}y + \frac{1}{2} \text { from } \frac{1}{3} - \frac{5}{3} y^2\]

Take away: 

\[\frac{2}{3}ac - \frac{5}{7}ab + \frac{2}{3}bc\text { from } \frac{3}{2}ab - \frac{7}{4}ac - \frac{5}{6}bc\]

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 − x² + 5 and − 4x − 3 + 7x² from 5.

Simplify the following:

 x² − 3x + 5 −  \[\frac{1}{2}\] (3x² − 5x + 7)

Simplify the following:

 [5 − 3x + 2y − (2x − y)] − (3x − 7y + 9)

Simplify the following:

\[\frac{11}{2} x^2 y - \frac{9}{4}x y^2 + \frac{1}{4}xy - \frac{1}{14} y^2 x + \frac{1}{15}y x^2 + \frac{1}{2}xy\]

Simplify the following:

\[\left( \frac{1}{3} y^2 - \frac{4}{7}y + 11 \right) - \left( \frac{1}{7}y - 3 + 2 y^2 \right) - \left( \frac{2}{7}y - \frac{2}{3} y^2 + 2 \right)\]

Simplify the following: \[- \frac{1}{2} a^2 b^2 c + \frac{1}{3}a b^2 c - \frac{1}{4}ab c^2 - \frac{1}{5}c b^2 a^2 + \frac{1}{6}c b^2 a - \frac{1}{7} c^2 ab + \frac{1}{8}c a^2 b .\]

Find each of the following product:
5x² × 4x³

Find each of the following product:
−3a² × 4b⁴

Find each of the following product:
(−5xy) × (−3x²yz)

Find each of the following product: 

\[\frac{1}{4}xy \times \frac{2}{3} x^2 y z^2\]

Find each of the following product: 

\[\left( - \frac{7}{5}x y^2 z \right) \times \left( \frac{13}{3} x^2 y z^2 \right)\]

Find each of the following product: \[\left( \frac{- 24}{25} x^3 z \right) \times \left( - \frac{15}{16}x z^2 y \right)\]

Find each of the following product:

\[\left( - \frac{1}{27} a^2 b^2 \right) \times \left( \frac{9}{2} a^3 b^2 c^2 \right)\]

Find each of the following product: \[( - 7xy) \times \left( \frac{1}{4} x^2 yz \right)\]

Find each of the following product:
(7ab) × (−5ab²c) × (6abc²)

Find each of the following product:
(−5a) × (−10a²) × (−2a³)

Find each of the following product:
(−4x²) × (−6xy²) × (−3yz²)

Find each of the following product:

\[\left( - \frac{2}{7} a^4 \right) \times \left( - \frac{3}{4} a^2 b \right) \times \left( - \frac{14}{5} b^2 \right)\]

Find each of the following product: \[\left( \frac{7}{9}a b^2 \right) \times \left( \frac{15}{7}a c^2 b \right) \times \left( - \frac{3}{5} a^2 c \right)\]

Find each of the following product: \[\left( \frac{4}{3} u^2 vw \right) \times \left( - 5uv w^2 \right) \times \left( \frac{1}{3} v^2 wu \right)\]

Find each of the following product:

\[\left( 0 . 5x \right) \times \left( \frac{1}{3}x y^2 z^4 \right) \times \left( 24 x^2 yz \right)\]

Find each of the following product: \[\left( \frac{4}{3}p q^2 \right) \times \left( - \frac{1}{4} p^2 r \right) \times \left( 16 p^2 q^2 r^2 \right)\]

Find each of the following product:
(2.3xy) × (0.1x) × (0.16)

Express each of the following product as a monomials and verify the result in each case for x = 1:
(3x) × (4x) × (−5x)

Express each of the following product as a monomials and verify the result in each case for x = 1:
(4x²) × (−3x) × \[\left( \frac{4}{5} x^3 \right)\]

Express each of the following product as a monomials and verify the result in each case for x = 1:
(5x⁴) × (x²)³ × (2x)²

Express each of the following product as a monomials and verify the result in each case for x = 1:
(x²)³ × (2x) × (−4x) × (5)

Write down the product of −8x²y⁶ and −20xy. Verify the product for x = 2.5, y = 1.

Evaluate (3.2x⁶y³) × (2.1x²y²) when x = 1 and y = 0.5.

Find the value of (5x⁶) × (−1.5x²y³) × (−12xy²) when x = 1, y = 0.5.

Evaluate (2.3a⁵b²) × (1.2a²b²) when a = 1 and b = 0.5.

Evaluate (−8x²y⁶) × (−20xy) for x = 2.5 and y = 1.

Express each of the following product as a monomials and verify the result for x = 1, y = 2:
(−xy³) × (yx³) × (xy)

Express each of the following product as a monomials and verify the result for x = 1, y = 2: \[\left( \frac{1}{8} x^2 y^4 \right) \times \left( \frac{1}{4} x^4 y^2 \right) \times \left( xy \right) \times 5\]

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

Express each of the following product as a monomials and verify the result for x = 1, y = 2: \[\left( - \frac{4}{7} a^2 b \right) \times \left( - \frac{2}{3} b^2 c \right) \times \left( - \frac{7}{6} c^2 a \right)\]

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

\[\left( \frac{4}{9}ab c^3 \right) \times \left( - \frac{27}{5} a^3 b^2 \right) \times \left( - 8 b^3 c \right)\]

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

\[(2xy) \times \left( \frac{x^2 y}{4} \right) \times \left( x^2 \right) \times \left( y^2 \right)\]

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

\[\left( \frac{3}{5} x^2 y \right) \times \left( - \frac{15}{4}x y^2 \right) \times \left( \frac{7}{9} x^2 y^2 \right)\]

Find the following product:
2a³(3a + 5b)

Find the following product:
−11a(3a + 2b)

Find the following product:
−5a(7a − 2b)

Find the following product:
−11y²(3y + 7)

Find the following product: \[\frac{6x}{5}( x^3 + y^3 )\]

xy(x³ − y³)

Find the following product:
0.1y(0.1x⁵ + 0.1y)

Find the following product: \[\left( - \frac{7}{4}a b^2 c - \frac{6}{25} a^2 c^2 \right)( - 50 a^2 b^2 c^2 )\]

Find the following product: \[- \frac{8}{27}xyz\left( \frac{3}{2}xy z^2 - \frac{9}{4}x y^2 z^3 \right)\]

Find the following product: \[- \frac{4}{27}xyz\left( \frac{9}{2} x^2 yz - \frac{3}{4}xy z^2 \right)\]

Find the following product:
1.5x(10x²y − 100xy²)

Find the following product:
4.1xy(1.1x − y)

Find the following product:
250.5xy \[\left( xz + \frac{y}{10} \right)\]

Find the following product: \[\frac{7}{5} x^2 y\left( \frac{3}{5}x y^2 + \frac{2}{5}x \right)\]

Find the following product: \[\frac{4}{3}a( a^2 + b^2 - 3 c^2 )\]

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

Find the product −3y(xy + y²) and find its value for x = 4 and y = 5.

Multiply \[- \frac{3}{2} x^2 y^3 by (2x - y)\] and verify the answer for x = 1 and y = 2.

Multiply the monomial by the binomial and find the value for x = −1, y = 0.25 and z = 0.05: 15y²(2 − 3x)

Multiply the monomial by the binomial and find the value for x = −1, y = 0.25 and z = 0.05: −3x(y² + z²)

Multiply the monomial by the binomial and find the value for x = −1, y = 0.25 and z = 0.05: z²(x − y)

Multiply the monomial by the binomial and find the value for x = −1, y = 0.25 and z = 0.05:

xz(x² + y²)

Simplify: 2x²(x³ − x) − 3x(x⁴ + 2x) − 2(x⁴ − 3x²)

Simplify: x³y(x² − 2x) + 2xy(x³ − x⁴)

Simplify: 3a² + 2(a + 2) − 3a(2a + 1)

Simplify: x(x + 4) + 3x(2x² − 1) + 4x² + 4

Simplify:  a(b − c) − b(c − a) − c(a − b)

Simplify: a(b − c) + b(c − a) + c(a − b)

Simplify: 4ab(a − b) − 6a²(b − b²) − 3b²(2a² − a) + 2ab(b − a)

Simplify:  x²(x² + 1) − x³(x + 1) − x(x³ − x)

Simplify:   2a² + 3a(1 − 2a³) + a(a + 1)

Simplify: a²(2a − 1) + 3a + a³ − 8

Simplify: \[\frac{3}{2} x^2 ( x^2 - 1) + \frac{1}{4} x^2 ( x^2 + x) - \frac{3}{4}x( x^3 - 1)\]

Simplify: a²b(a − b²⁾ + ab²(4ab − 2a²) − a³b(1 − 2b)

Simplify:  a²b(a³ − a + 1) − ab(a⁴ − 2a² + 2a) − b (a³ − a² − 1)

Multiply:
(5x + 3) by (7x + 2)

Multiply:
(2x + 8) by (x − 3)

Multiply:
(7x + y) by (x + 5y)

Multiply:
(a − 1) by (0.1a² + 3)

Multiply:
(3x² + y²) by (2x² + 3y²)

Multiply: \[\left( \frac{3}{5}x + \frac{1}{2}y \right) by \left( \frac{5}{6}x + 4y \right)\]

Multiply:
(x⁶ − y⁶) by (x² + y²)

Multiply:
(x² + y²) by (3a + 2b)

Multiply:
[−3d + (−7f)] by (5d + f)

Multiply:
(0.8a − 0.5b) by (1.5a − 3b)

Multiply:
(2x²y² − 5xy²) by (x² − y²)

Multiply: \[\left( \frac{x}{7} + \frac{x^2}{2} \right)by\left( \frac{2}{5} + \frac{9x}{4} \right)\]

Multiply: \[\left( - \frac{a}{7} + \frac{a^2}{9} \right)by\left( \frac{b}{2} - \frac{b^2}{3} \right)\].

Multiply:
(3x²y − 5xy²) by  \[\left( \frac{1}{5} x^2 + \frac{1}{3} y^2 \right)\].

Multiply:
(2x² − 1) by (4x³ + 5x²)

(2xy + 3y²) (3y² − 2)

Find the following product and verify the result for x = − 1, y = − 2:
(3x − 5y) (x + y)

Find the following product and verify the result for x = − 1, y = − 2:
(x²y − 1) (3 − 2x²y)

Find the following product and verify the result for x = − 1, y = − 2: \[\left( \frac{1}{3}x - \frac{y^2}{5} \right)\left( \frac{1}{3}x + \frac{y^2}{5} \right)\]

Simplify:
x²(x + 2y) (x − 3y)

Simplify:

(x² − 2y²) (x + 4y) x²y²

Simplify:
a²b²(a + 2b)(3a + b)

Simplify:
x²(x − y) y²(x + 2y)

Simplify:
(x³ − 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:
(x³ − 2x² + 3x − 4) (x −1) − (2x − 3)(x² − x + 1)

Write the following square of binomial as trinomial: (x + 2)²

Write the following square of binomial as trinomial:  (8a + 3b)²

Write the following square of binomial as trinomial:  (2m + 1)²

Write the following square of binomial as trinomial:  \[\left( 9a + \frac{1}{6} \right)^2\]

Write the following square of binomial as trinomial: 

\[\left( x + \frac{x^2}{2} \right)^2\]

Write the following square of binomial as trinomial:  \[\left( \frac{x}{4} - \frac{y}{3} \right)\]

Write the following square of binomial as trinomial:  \[\left( 3x - \frac{1}{3x} \right)^2\]

Write the following square of binomial as trinomial:  \[\left( \frac{x}{y} - \frac{y}{x} \right)^2\]

Write the following square of binomial as trinomial: \[\left( \frac{3a}{2} - \frac{5b}{4} \right)^2\]

Write the following square of binomial as trinomial: (a²b − bc²)²

Write the following square of binomial as trinomial:  \[\left( \frac{2a}{3b} + \frac{2b}{3a} \right)^2\]

Write the following square of binomial as trinomial:  (x² − ay)²

Find the product of the following binomial: (2x + y)(2x + y)

Find the product of the following binomial:  (a + 2b)(a − 2b)

Find the product of the following binomial:  (a² + bc)(a² − bc)

Find the product of the following binomial: \[\left( \frac{4x}{5} - \frac{3y}{4} \right)\left( \frac{4x}{5} + \frac{3y}{4} \right)\]

Find the product of the following binomial: \[\left( 2x + \frac{3}{y} \right)\left( 2x - \frac{3}{y} \right)\]

Find the product of the following binomial: (2a³ + b³)(2a³ − b³)

Find the product of the following binomial: \[\left( x^4 + \frac{2}{x^2} \right)\left( x^4 - \frac{2}{x^2} \right)\]

Find the product of the following binomial: \[\left( x^3 + \frac{1}{x^3} \right)\left( x^3 - \frac{1}{x^3} \right)\]

Using the formula for squaring a binomial, evaluate the following:  (102)²

Using the formula for squaring a binomial, evaluate the following: (99)²

Using the formula for squaring a binomial, evaluate the following:  (1001)²

Using the formula for squaring a binomial, evaluate the following:  (999)²

Using the formula for squaring a binomial, evaluate the following:  (703)²

Simplify the following using the formula: (a − b)(a + b) = a² − b²: (82)² − (18)²

Simplify the following using the formula: (a − b)(a + b) = a² − b²: (467)² − (33)²

Simplify the following using the formula: (a − b)(a + b) = a² − b²: (79)² − (69)²

Simplify the following using the formula: (a − b)(a + b) = a² − b²: 197 × 203

Simplify the following using the formula: (a − b)(a + b) = a² − b²: 113 × 87

Simplify the following using the formula: (a − b)(a + b) = a² − b²: 95 × 105

Simplify the following using the formula: (a − b)(a + b) = a² − b²: 1.8 × 2.2

Simplify the following using the formula: (a − b)(a + b) = a² − b²:  9.8 × 10.2

Simplify the following using the identities: \[\frac{{58}^2 - {42}^2}{16}\]

Simplify the following using the identities: 178 × 178 − 22 × 22

Simplify the following using the identities: \[\frac{198 \times 198 - 102 \times 102}{96}\]

Simplify the following using the identities:  1.73 × 1.73 − 0.27 × 0.27

Simplify the following using the identities: \[\frac{8 . 63 \times 8 . 63 - 1 . 37 \times 1 . 37}{0 . 726}\]

Find the value of x, if 4x = (52)² − (48)².

Find the value of x, if 14x = (47)² − (33)².

Find the value of x, if 5x = (50)² − (40)².

If \[x + \frac{1}{x} = 20,\]find the value of \[x^2 + \frac{1}{x^2} .\].

If \[x - \frac{1}{x} = 3,\]  find the values of \[x^2 + \frac{1}{x^2}\] and \[x^4 + \frac{1}{x^4} .\]

If \[x^2 + \frac{1}{x^2} = 18,\]  find the values of \[x + \frac{1}{x} \text { and } x - \frac{1}{x} .\]

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 value of the following expression: 16x² + 24x + 9, when \[x = \frac{7}{4}\]

Find the value of the following expression:  64x² + 81y² + 144xy, when x = 11 and \[y = \frac{4}{3}\]

Find the value of the following expression:  81x² + 16y² − 72xy, when \[x = \frac{2}{3}\] and  \[y = \frac{3}{4}\]

If \[x + \frac{1}{x} = 9,\]  find the value of \[x^4 + \frac{1}{x^4} .\]

If \[x + \frac{1}{x} = 12,\]  find the value of \[x - \frac{1}{x} .\]

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

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

If x² + y² = 29 and xy = 2, find the value of x - y.

If x² + y² = 29 and xy = 2, find the value of x⁴ + y⁴ .

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

4x² − 12x + 7

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

4x² − 20x + 20

Simplify : (x − y)(x + y) (x² + y²)(x⁴ + y²)

Simplify : (2x − 1)(2x + 1)(4x² + 1)(16x⁴ + 1)

Simplify : (4m − 8n)² + (7m + 8n)²

Simplify : (2.5p − 1.5q)² − (1.5p − 2.5q)²

Simplify :  (m² − n²m)² + 2m³n²

Show that:  (3x + 7)² − 84x = (3x − 7)²

Show that:  (9a − 5b)² + 180ab = (9a + 5b)² 

Show that: \[\left( \frac{4m}{3} - \frac{3n}{4} \right)^2 + 2mn = \frac{16 m^2}{9} + \frac{9 n^2}{16}\]

Show that:  (4pq + 3q)² − (4pq − 3q)² = 48pq²

Show that:  (a − b)(a + b) + (b − c)(b + c) + (c − a)( c + a) = 0

Find the following product: (x + 4) (x + 7)

Find the following product:  (x − 11) (x + 4)

Find the following product: (x + 7) (x − 5)

Find the following product: (x − 3) ( x − 2)

Find the following product: (y² − 4) (y² − 3)

Find the following product: \[\left( x + \frac{4}{3} \right)\left( x + \frac{3}{4} \right)\]

Find the following product: (3x + 5) (3x + 11)

Find the following product:  (2x² − 3) (2x² + 5)

Find the following product: (z² + 2) (z² − 3)

Find the following product: (3x − 4y) (2x − 4y)

Find the following product:  (3x² − 4xy) (3x² − 3xy)

Find the following product: \[\left( x + \frac{1}{5} \right)(x + 5)\]

Find the following product: \[\left( z + \frac{3}{4} \right)\left( z + \frac{4}{3} \right)\]

Find the following product:  (x² + 4) (x² + 9)

Find the following product: (y² + 12) (y² + 6)

Find the following product: \[\left( y^2 + \frac{5}{7} \right)\left( y^2 - \frac{14}{5} \right)\]

Find the following product:  (p² + 16) \[\left( p^2 - \frac{1}{4} \right)\]

Evaluate the following: 102 × 106

Evaluate the following: 109 × 107

Evaluate the following: 35 × 37

Evaluate the following: 53 × 55

Evaluate the following: 103 × 96

Evaluate the following: 34 × 36

Evaluate the following: 994 × 1006