**Algebraic Expressions and Identities**

**Question 1.****Find the product of the following pairs of monomials :(i)** 4, 7p

**(ii)**-4p, 7p

**(iii)**– 4p, 7pq

**(iv)**4p

^{3}, – 3p

**(v)**4p, 0

**Solution:****(i)** 4 x 7p = (4 x 7) x p = 28p**(ii)** – 4 p x 7p = (- 4 x 7) x (px P)

= -28p^{1} + ^{1} = – 28p^{2}**(iii)** -4px 7pq = (- 4 x 7) x (p x p x q)

= -28 p^{1+1}q = – 28p^{2}q**(iv)** 4p^{3} x – 3p = (4 x – 3) x (p^{3} x p)

= – 12p^{3+1} = =-12p^{4}**(v)** 4p x 0 = (4 x 0) x p = 0 x p = 0

**Question 2.****Find the areas of rectangles with the following pairs of monomials as their lengths and breadths respectively :**(p, q); (10m, 5n); (20x

^{2}, 5y

^{2}); (4x, 3x

^{2}); (3mn, 4np)

**Solution:**

We know that the area of a rectangle = l x b, where l = length and b = breadth.

Therefore, the areas of rectangles with pair of monomials (p, q); (10m, 5n); (20x^{2}, 5y^{2}); (4x, 3x^{2}) and (3mn, 4np) as their lengths and breadths are given by

pxq=pq

10 m x 5n = (10 x 5) x (m x n) = 50 mn

20x^{2} x 5y^{2} = (20 x 5) x (x^{2} x y^{2}) = 100x^{2}y^{2}

4x x 3x^{2} = (4 x 3) x (x x x^{2})

= 12x^{2}

and, 3 mn x 4np = (3×4 )x(mxnxnxp)

=12 mn^{2}p

**Question 3.****Complete the table of products :**

**Solution:**

Completed table is as under :

**Question 4.****Obtain the volume of rectangular boxes with the following length, breadth and height respectively(i)** 5a, 3a

^{2}, 7a

^{4}

**(ii)**2p, 4q, 8r

**(iii)**xy, 2x

^{2}y, 2xy

^{2}

**(iv)**a, 2b, 3c

**Solution:****(i)** Required volume = 5a x 3a^{2} x 7a^{4}

= (5 x 3 x 7) x (a x a^{2} x a^{4})

= 105a^{1+2+4} =105a^{7}

**(ii)** Required volume = 2p x 4q x 8r

= (2 x 4 x 8 )x p x q x r = 64 pqr

**(iii)** Required volume =xy x 2x^{2}y x 2xy^{2}

= (1 x 2 x 2) x (x x x^{2} x x x y x y x y^{2})

= 4x^{1+2+1} y^{1+1+2} = 4x^{4}y^{4}

**(iv)** Required volume =a x 2b x 3c

= (1 x 2 x 3) x (a x b x c)

= 6abc

**Question 5.****Obtain the product of(i)** xy, yz, zx

**(ii)**a, – a

^{2}, a

^{3}

**(iii)**2, 4y, 8y

^{2}, 16y

^{3}

**(iv)**a, 26, 3c, 6a6c

**(v)**m, – mn, mnp

**Solution:**