Ex.9.5 Q4 Algebraic Expressions and Identities Solution - NCERT Maths Class 8

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Question

Simplify.

(i) \( ({a}^{2}-{b}^{2} )^{2} \)

(ii) \(\begin{align}{{\left( 2x+5 \right)}^{2}}-{{\left( 2x-5 \right)}^{2}}\end{align}\)

(iii)\(\begin{align}{{\left( 7m-8n \right)}^{2}}+{{\left( 7m+8n \right)}^{2}}\end{align}\)

(iv)\(\begin{align}{{\left( 4m+5n \right)}^{2}}+{{\left( 5m+4n \right)}^{2}}\end{align}\)

(v)\(\begin{align}{{\left( 2.5p-1.5q \right)}^{2}}-{{\left( 1.5p-2.5q \right)}^{2}}\end{align}\)

(vi)\(\begin{align}{{\left( ab+bc \right)}^{2}}-2a{{b}^{2}}c\end{align}\)

(vii)\(\begin{align}{{\left( {{m}^{2}}-{{n}^{2}}m \right)}^{2}}+2{{m}^{3}}{{n}^{2}}\end{align}\)

Text Solution

What is known?

Expressions

What is unknown?

Simplification

Reasoning:

\[\begin{align}{(a + b)^2} &= {a^2} + 2ab + {b^2}\\{(a - b)^2} &= {a^2} - 2ab + {b^2}\\(a + b)(a - b) &= {a^2} - {b^2}\end{align}\]

Steps:

(i)

\[\begin{align}({a}^{2}-{b}^{2} )^{2}&= {\left( {{a^2}} \right)^2} - 2\left( {{a^2}} \right)\left( {{b^2}} \right) + {\left( {{b^2}} \right)^2} \\{[{\left( {a - b} \right)}^2} &= {a^2} - 2ab + {b^2}]\\&= {a^4} - 2{a^2}{b^2} + {b^4}\end{align}\]

(ii) \(\begin{align} \quad {{\left( 2x+5 \right)}^{2}}-{{\left( 2x-5 \right)}^{2}}\end{align}\)

\[\begin{align}&=\begin{Bmatrix}  {\left( {2x} \right)^2} + 2\left( {2x} \right)\left( 5 \right) + {\left( 5 \right)^2} -\\ [ \left( {2x} \right)^2 - 2\left( {2x} \right)\left( 5 \right)  + {{\left( 5 \right)}^2}] \end{Bmatrix}  \\&\qquad \left[ {{{\left( {a - b} \right)}^2} = {a^2} - 2ab + {b^2}} \right]\\&\qquad\left[ {{{\left( {a + b} \right)}^2} = {a^2} + 2ab + {b^2}} \right]\\& =\begin{Bmatrix} 4{x^2} + 20x + 25 \\ - \left[ {4{x^2} - 20x + 25} \right] \end{Bmatrix} \\&= \begin{Bmatrix} \not\!4{x^2} + \not\!\!20x + 25 \\ -\not\!4 {x^2} + \not\!\!20x - 25 \end{Bmatrix} \\& = 40x\end{align}\]

(iv)\(\begin{align}{{\left( 4m+5n \right)}^{2}}+{{\left( 5m+4n \right)}^{2}}\end{align}\)

\[\begin{align}&= \begin{Bmatrix} {\left( {4m} \right)^2} + 2\left( {4m} \right)\left( {5n} \right) \\ + {\left( {5n} \right)^2} + {\left( {5m} \right)^2} \\ + 2\left( {5m} \right)\left( {4n} \right) + {\left( {4n} \right)^2}\end{Bmatrix} \\ & \quad  \left[ {{{\left( {a + b} \right)}^2} = {a^2} + 2ab + {b^2}} \right]\\&=\begin{Bmatrix} 16{m^2} + 40mn \\ + 25{n^2} + 25{m^2} \\ + 40mn + 16{n^2} \end{Bmatrix}\\&= 41{m^2} + 80mn + 41{n^2}\end{align}\]