Exercise 9.2 Algebraic Expressions and Identities- NCERT Solutions Class 8

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Question 1

Find the product of the following pairs of monomials.

(i) \(\quad 4,7p\)

(ii)\(\quad -\text{ }4p,\text{ }7p~\)

(iii)\(\quad - 4p,\;7pq\)

(iv)\(\quad 4{p^3},\; - {\rm{ }}3p\)

(v)\(\quad 4p,\;0\)

Solution

Video Solution

What is known?

Pairs of monomials

What is unknown?

Product

Reasoning:

i) By using the distributive law, we can carry out the multiplication term by term.

ii) In multiplication of polynomials with polynomials, we should always look for like terms, if any, and combine them.

Steps:

The product will be as follows.

(i)

\[\begin{align} &4 \times 7p\\ &= 4 \times 7 \times p \\&= 28p \end{align}\]

(ii)

\[\begin{align} & - 4p \times 7p \\&=- 4 \times p \times 7 \times p \\&= \left( { - 4 \times 7} \right) \times \left( {p \times p} \right) \\&=- 28{p^2} \end{align}\]

(iii)

\[\begin{align} &  - 4p \times 7pq \\&=- 4 \times p \times 7 \times p \times q \\&= \left( { - 4 \times 7} \right) \times \left( {p \times p \times q} \right) \\& =- 28{p^2}q \end{align}\]

(iv)

\[\begin{align} &  4{p^3} \times- 3p \\&= 4 \times \left( { - 3} \right) \times p \times p \times p \times p \\&=- 12{p^4} \end{align}\]

(v)

\[\begin{align} & 4p \times 0 \\&= 4 \times p \times 0 \\&= 0 \end{align}\]

Question 2

Find the areas of rectangles with the following pairs of monomials as their lengths and breadths respectively.

(i) \((p,\,q)\)

(ii) \(\left( 10m,\,5n \right)\)

(iii) \(\left( 20 x^2,\,5 y^2  \right)\)

(iv) \(\left( 4x,\,3x^2 \right)\)

(v) \(\left( 3mn,\,4np \right)\)

Solution

Video Solution

What is known?

Lengths and breadths of rectangles

What is unknown?

Areas of rectangles

Reasoning:

\[\begin{align} &{\text{Area of a Rectangle}} \\&= \text{Length}\,\times\, \text{Breadth}\end{align} \]

Steps:

We know that,

\[\begin{align} &{\text{Area of a Rectangle}} \\&= \text{Length}\,\times\, \text{Breadth}\end{align} \]

(i) Area of \(1^{\rm st}\) rectangle 

\[\begin{align} &=p\,\times\,q\\ &= pq\end{align}\]

(ii) Area of \(2^{\rm nd} \) rectangle

\[\begin{align}& = 10m \times5n\\&= 10\times5\times m \times n\\& = 50\,mn \end{align}\]

(iii) Area of \(3^{\rm rd}\) rectangle

\[\begin{align}  &= 20{x^2} \times 5{y^2}\\&= {\rm{ }}20{\rm{ }} \times 5\times {x^2} \times {y^2} \\&= 100{x^2}{y^2} \end{align}\]

(iv) Area of \(4^{\rm th}\)  rectangle

\[\begin{align} & =4x \times 3{x^2}\\&= 4 \times3 \times x \times {x^2} \\&= 12{x^3} \end{align}\]

(v) Area of \( 5^{\rm th}\) rectangle

\[\begin{align}  &= 3mn\times 4np\\ &= 3 \times 4 \times m \times n \times n \times p \\&= 12m{n^2}p\end{align}\]

Question 3

Complete the table of products.

\(\begin{align}\frac{{{\rm{First}}\,{\rm{monomial}} \to }}{{{\rm{Second}}\,{\rm{monomial}} \downarrow }}\end{align}\) \(2x\) \( - 5y\) \(3{x^2}\) \( - 4xy\) \(7{x^2}y\) \( - 9{x^2}{y^2}\)
\(2x\) \(4{x^2}\) \( \dots \) \( \dots \) \( \dots \) \( \dots \) \( \dots \)
\( - 5y\) \( \dots \) \( \dots \) \( - 15{x^2}y\) \( \dots \) \( \dots \) \( \dots \)
\(3{x^2}\) \( \dots \) \( \dots \) \( \dots \) \( \dots \) \( \dots \) \( \dots \)
\( - 4xy\) \( \dots \) \( \dots \) \( \dots \) \( \dots \) \( \dots \) \( \dots \)
\(7{x^2}y\) \( \dots \) \( \dots \) \( \dots \) \( \dots \) \( \dots \) \( \dots \)
\( - 9{x^2}{y^2}\) \( \dots \) \( \dots \) \( \dots \) \( \dots \) \( \dots \) \( \dots \)

Solution

Video Solution

What is known?

Expressions

What is Unknown?

Product

Steps:

The table can be completed as follows.

\(\begin{align}\frac{{{\rm{First}}\,{\rm{monomial}} \to }}{{{\rm{Second}}\,{\rm{monomial}} \downarrow }}\end{align}\) \(2x\) \( - 5y\) \(3{x^2}\) \( - 4xy\) \(7{x^2}y\) \( - 9{x^2}{y^2}\)
\(2x\) \(4{x^2}\) \( - 10xy\) \(6{x^3}\) \( - 8{x^2}y\) \(14{x^3}y\) \( - 18{x^3}{y^2}\)
\( - 5y\) \( - 10xy\) \(25{y^2}\) \( - 15{x^2}y\) \(20x{y^2}\) \( - 35{x^2}{y^2}\) \(45{x^2}{y^3}\)
\(3{x^2}\) \(6{x^3}\) \( - 15{x^2}y\) \(9{x^4}\) \( - 12{x^3}y\) \(21{x^4}y\) \( - 27{x^4}{y^2}\)
\( - 4xy\) \( - 8{x^2}y\) \(20x{y^2}\) \( - 12{x^3}y\) \(16{x^2}{y^2}\) \( - 28{x^3}{y^2}\) \(36{x^3}{y^3}\)
\(7{x^2}y\) \(14{x^3}y\) \( - 35{x^2}{y^2}\) \(21{x^4}y\) \( - 28{x^3}{y^2}\) \(49{x^4}{y^2}\) \( - 63{x^4}{y^3}\)
\( - 9{x^2}{y^2}\) \( - 18{x^3}{y^2}\) \(45{x^2}{y^3}\) \( - 27{x^4}{y^2}\) \(36{x^3}{y^3}\) \( - 63{x^4}{y^3}\) \(81{x^4}{y^4}\)

Question 4

Obtain the volume of rectangular boxes with the following length, breadth and height respectively.

(i) \(\quad 5a,\,3{a^2},\,7{a^4}\)

(ii) \(\quad 2p,4q,8r\)

(iii) \( \quad xy,\,2{x^2}y,\,2x{y^2}\)

(iv) \(\quad a,\,2b,\,3c\)

Solution

Video Solution

What is known?

Length, breadth and height respectively of rectangular boxes

What is unknown?

Volume of rectangular boxes

Reasoning:

Volume of a Rectangular Box

 \(=\)  Length \(\times\)  Breadth \(\times\) Height

Steps:

We know that,

Volume of a Rectangular Box

 \(=\)  Length \(\times\)  Breadth \(\times\) Height

(i)

\(\begin{align}& {\rm{Volume }} \\&= 5a \times 3{a^2} \times 7{a^4} \\&= 5 \times 3 \times 7 \times a \times {a^2} \times {a^4} \\&= 105{a^7}\end{align}\)

(ii)

\(\begin{align}& {\rm{Volume }} \\ &= 2p \times 4q \times 8r \\ &= 2 \times 4 \times 8 \times p \times q \times r \\ &= 64pqr\end{align}\)

(iii)

\(\begin{align} &{\rm{ Volume }}\\& = xy \times 2{x^2}y \times 2x{y^2}\\ & = 2 \times 2 \times xy \times {x^2}y \times x{y^2} \\&= 4{x^4}{y^4}\end{align}\)

(iv)

\(\begin{align}& {\rm{Volume }} \\ &= a \times 2b \times 3c \\ &= 2 \times 3 \times a \times b \times c \\&= 6abc\end{align}\)

Question 5

Obtain the product of

(i) \(\quad xy,\,yz,\,zx\)

(ii)  \(\quad a,\, - {a^2},\,{a^3}\)

(iii)\(\quad 2,\,4y,\,8{y^2},\,16{y^3}\)

(iv)\(\quad a,\,2b,\,3c,\,6abc\)

(v) \(\quad m,\, - mn,\,mnp\)

Solution

Video Solution

What is known?

Expressions

What is unknown?

Product

Reasoning:

By using the distributive law, we can carry out the multiplication term by term.

Steps:

(i) 

\[\begin{align} xy \times yz \times zx = {x^2}{y^2}{z^2}\end{align}\]

(ii) 

\[\begin{align} a \times \left( { - {a^2}} \right) \times {a^3} = - {a^6}\end{align}\]

(iii) 

\[\begin{align} & 2 \times 4y \times 8{y^2} \times 16{y^3} \\&= 2 \times 4 \times 8 \times 16 \times y \times {y^2} \times {y^3} \\ &= 1024{y^6}\end{align}\]

(iv) 

\[\begin{align}& a \times 2b \times 3c \times 6abc \\ &= 2 \times 3 \times 6 \times a \times b \times c \times abc \\ &= 36{a^2}{b^2}{c^2}\end{align}\]

(v)

\[\begin{align} m \times \left( { - mn} \right) \times mnp = - {m^3}{n^2}p\end{align}\]

  
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