Question

Obtain the product of the following
 $ \left( i \right)xy,yz,zx $
 $ \left( {ii} \right)a, - {a^2},{a^3} $
 $ \left( {iii} \right)2,4y,8{y^2},16{y^3} $
 $ \left( {iv} \right)a,2b,3c,6abc $
 $ \left( v \right)m, - mn,mnp $

Answer 1

Hint: When terms with similar variables (exponent>0) irrespective of their exponents are multiplied, then the exponent of the result must be greater than the multiplicands. Anything multiplied with zero is zero.

Complete step-by-step answer:
 A monomial is a polynomial with just one term and a monomial is a number, a variable, a product of a number and a variable where all exponents are whole numbers. Any number, all by itself, can be a monomial, like the number 5 or the number 2,700. To add two or more monomials that are like terms, add the coefficients; keep the variables and exponents on the variables the same. To subtract two or more monomials that are like terms, subtract the coefficients; keep the variables and exponents on the variables the same.

 $ \left( i \right)xy,yz,zx $
Here, ‘xy’, ‘yz’, ‘zx’ are monomials with degree 2.
‘xy’ has variables ‘x’ and ‘y’.
‘yz’ has variables ‘y’ and ‘z’.
‘zx’ has variables ‘z’ and ‘x’.
 $
  xy \times yz \times zx = x \times y \times y \times z \times z \times x \\
   = {x^1} \times {y^1} \times {y^1} \times {z^1} \times {z^1} \times {x^1} \\
   = {x^1} \times {x^1} \times {y^1} \times {y^1} \times {z^1} \times {z^1} \\
   = {x^{1 + 1}} \times {y^{1 + 1}} \times {z^{1 + 1}} \\
  \left( {\because {a^m} \times {a^n} = {a^{m + n}}} \right) \\
   = {x^2} \times {y^2} \times {z^2} \\
   = {x^2}{y^2}{z^2} \\
  $
The product of xy, yz, zx is $ {x^2}{y^2}{z^2} $ .

 $ \left( {ii} \right)a, - {a^2},{a^3} $
Here $ a, - {a^2},{a^3} $ are monomials with degree 1, 2, 3 respectively and the three terms have a single variable ‘a’ .
 $
  a \times \left( { - {a^2}} \right) \times {a^3} = {a^1} \times \left( { - {a^2}} \right) \times {a^3} \\
   = - \left( {{a^1} \times {a^2} \times {a^3}} \right) \\
   = - \left( {{a^{1 + 2 + 3}}} \right) \\
  \left( {\because {a^m} \times {a^n} = {a^{m + n}}} \right) \\
   = - \left( {{a^6}} \right) \\
   = - {a^6} \\
  $
The product of $ a, - {a^2},{a^3} $ is $ - {a^6} $ .

 $ \left( {iii} \right)2,4y,8{y^2},16{y^3} $
Here ‘2’, $ 4y,8{y^2},16{y^3} $ are monomials with degree 0,1,2,3 respectively and have the same variable ‘y’.
 $
  2 \times 4y \times 8{y^2} \times 16{y^3} = 2 \times 4 \times y \times 8 \times {y^2} \times 16 \times {y^3} \\
   = 2 \times 4 \times 8 \times 16 \times {y^1} \times {y^2} \times {y^3} \\
   = 1024 \times {y^{1 + 2 + 3}} \\
  \left( {\because {a^m} \times {a^n} = {a^{m + n}}} \right) \\
   = 1024 \times {y^6} \\
   = 1024{y^6} \\
  $
The product of $ 2,4y,8{y^2},16{y^3} $ is $ 1024{y^6} $ .

 $ \left( {iv} \right)a,2b,3c,6abc $
Here $ a,2b,3c,6abc $ are monomials with degree 1,1,1,3 respectively and have variables a, b, c.
 $
  a \times 2b \times 3c \times 6abc = a \times 2 \times b \times 3 \times c \times 6 \times a \times b \times c \\
   = 2 \times 3 \times 6 \times {a^1} \times {a^1} \times {b^1} \times {b^1} \times {c^1} \times {c^1} \\
   = 36 \times {a^2} \times {b^2} \times {c^2} \\
   = 36{a^2}{b^2}{c^2} \\
  $
The product of $ a,2b,3c,6abc $ is $ 36{a^2}{b^2}{c^2} $ .
 $ \left( v \right)m, - mn,mnp $
Here $ m, - mn,mnp $ are monomials with degree 1, 2, 3 respectively and have variables m, n, p.
 $
  m \times \left( { - mn} \right) \times mnp = - \left( {m \times m \times n \times m \times n \times p} \right) \\
   = - \left( {{m^1} \times {m^1} \times {m^1} \times {n^1} \times {n^1} \times {p^1}} \right) \\
   = - \left( {{m^{1 + 1 + 1}} \times {n^{1 + 1}} \times {p^1}} \right) \\
  \left( {\because {a^m} \times {a^n} = {a^{m + n}}} \right) \\
   = - \left( {{m^3} \times {n^2} \times p} \right) \\
   = - {m^3}{n^2}p \\
  $
The product of $ m, - mn,mnp $ is $ - {m^3}{n^2}p $ .

Note: The degree of a monomial is defined as the sum of all the exponents of the variables, including the implicit exponents of 1 for the variables which appear without exponent. The degree of all the constants is zero.