Ex 9.5, 4 - Chapter 9 Class 8 Algebraic Expressions and Identities

Transcript

Ex 9.5, 4 Simplify. (i) (𝑎^2−𝑏^2 )^2 (𝑎^2−𝑏^2 )^2 (𝑥−𝑦)^2=𝑥^2+𝑦^2−2𝑥𝑦 Putting 𝑥 = 𝑎^2 & 𝑦 = 𝑏^2 = (𝑎^2 )^2+(𝑏^2 )^2−2(𝑎^2 )(𝑏^2 ) = 𝑎^(2 × 2)+𝑏^(2 × 2)−2𝑎^2 𝑏^2 = 𝒂^𝟒+𝒃^𝟒−𝟐𝒂^𝟐 𝒃^𝟐 Ex 9.5, 4 Simplify. (ii) (2𝑥+5)^2−(2𝑥−5)^2 Solving (𝟐𝒙+𝟓)^𝟐 (𝑎+𝑏)^2=𝑎^2+𝑏^2+2𝑎𝑏 Putting 𝑎 = 2𝑥 & 𝑏 = 5 = (2𝑥)^2+(5)^2+2(2𝑥)(5) = 4𝑥^2+25+20𝑥 Solving (𝟐𝒙−𝟓)^𝟐 (𝑎−𝑏)^2=𝑎^2+𝑏^2−2𝑎𝑏 Putting 𝑎 = 2𝑥 & 𝑏 = 5 = (2𝑥)^2+(5)^2−2(2𝑥)(5) = 4𝑥^2+25−20𝑥 = 4𝑥^2+25+20𝑥−4𝑥^2−25+20𝑥 = (4𝑥^2−4𝑥^2 )+(25−25)+(20𝑥+20𝑥) = 0+0+40𝑥 = 𝟒𝟎𝒙 Ex 9.5, 4 Simplify. (iii) (7𝑚−8𝑛)^2+(7𝑚+8𝑛)^2 Solving (𝟕𝒎−𝟖𝒏)^𝟐 (𝑎−𝑏)^2=𝑎^2+𝑏^2−2𝑎𝑏 Putting 𝑎 = 7𝑚 & 𝑏 = 8𝑛 = (7𝑚)^2+(8𝑛)^2−2(7𝑚)(8𝑛) = 49𝑚^2+64𝑛^2−(14×8)𝑚𝑛 = 49𝑚^2+64𝑛^2−112𝑚𝑛 Solving (𝟕𝒎+𝟖𝒏)^𝟐 (𝑎+𝑏)^2=𝑎^2+𝑏^2+2𝑎𝑏 Putting 𝑎 = 7𝑚 & 𝑏 = 8𝑛 = (7𝑚)^2+(8𝑛)^2+2(7𝑚)(8𝑛) = 49𝑚^2+64𝑛^2+(14×8)𝑚𝑛 = 49𝑚^2+64𝑛^2+112𝑚𝑛 ∴ (7𝑚−8𝑛)^2+(7𝑚+8𝑛)^2 = (49𝑚^2+64𝑛^2−112𝑚𝑛)+(49𝑚^2+64𝑛^2+112𝑚𝑛) = (49𝑚^2+49𝑚^2 )+(64𝑛^2+64𝑛^2 )+(−112𝑚𝑛+112𝑚𝑛) = 𝟗𝟖𝒎^𝟐+𝟏𝟐𝟖𝒏^𝟐 Ex 9.5, 4 Simplify. (iv) (4𝑚+5𝑛)^2+(5𝑚+4𝑛)^2 Solving (𝟒𝒎+𝟓𝒏)^𝟐 (𝑎+𝑏)^2=𝑎^2+𝑏^2+2𝑎𝑏 Putting 𝑎 = 4𝑚 & 𝑏 = 5𝑛 = (4𝑚)^2+(5𝑛)^2+2(4𝑚)(5𝑛) = 16𝑚^2+25𝑛^2+40𝑚𝑛 Solving (𝟓𝒎+𝟒𝒏)^𝟐 (𝑎+𝑏)^2=𝑎^2+𝑏^2+2𝑎𝑏 Putting 𝑎 = 5𝑚 & 𝑏 = 4𝑛 = (5𝑚)^2+(4𝑛)^2+2(5𝑚)(4𝑛) = 25𝑚^2+16𝑛^2+40𝑚𝑛 ∴ (4𝑚+5𝑛)^2+(5𝑚+4𝑛)^2 = (16𝑚^2+25𝑛^2+40𝑚𝑛)+(25𝑚^2+16𝑛^2+40𝑚𝑛) = (16𝑚^2+25𝑚^2 )+(25𝑛^2+16𝑛^2 )+(40𝑚𝑛+40𝑚𝑛) = 𝟒𝟏𝒎^𝟐+𝟒𝟏𝒏^𝟐+𝟖𝟎𝒎𝒏 Ex 9.5, 4 Simplify. (v) (2.5𝑝−1.5𝑞)^2 −(1.5𝑝−2.5𝑞)^2 Solving (𝟐.𝟓𝒑−𝟏.𝟓𝒒)^𝟐 (𝑎−𝑏)^2=𝑎^2+𝑏^2−2𝑎𝑏 Putting 𝑎 = 2.5𝑝 & 𝑏 = 1.5𝑞 = (2.5𝑝)^2+(1.5𝑞)^2− 2(2.5𝑝)(1.5𝑞) = (〖2.5〗^2×𝑝^2 )+(〖1.5〗^2×𝑞)^2 − (2×2.5×1.5)×(𝑝×𝑞) = (25/10)^2 𝑝^2+(15/10)^2 𝑞^2 − (2×25/10×15/10)𝑝𝑞 = 625/100 𝑝^2+225/100 𝑞^2−750/100 𝑝𝑞 Solving (𝟏.𝟓𝒑−𝟐.𝟓𝒒)^𝟐 (𝑎−𝑏)^2=𝑎^2+𝑏^2−2𝑎𝑏 Putting 𝑎 = 1.5𝑝 & 𝑏 = 2.5𝑞 = (1.5𝑝)^2+(2.5𝑞)^2−2(1.5𝑝)(2.5𝑞) = (〖1.5〗^2×𝑝^2 )+(〖2.5〗^2×𝑞)^2−(2×1.5×2.5)×(𝑝×𝑞) = (15/10)^2 𝑝^2+(25/10)^2 𝑞^2−(2×15/10×25/10)𝑝𝑞 = 225/100 𝑝^2+625/100 𝑞^2−750/100 𝑝𝑞 ∴ (2.5𝑝−1.5𝑞)^2 −(1.5𝑝−2.5𝑞)^2 = 625/100 𝑝^2+225/100 𝑞^2−750/100 𝑝𝑞−(225/100 𝑝^2+625/100 𝑞^2−750/100 𝑝𝑞" " ) = 625/100 𝑝^2+225/100 𝑞^2−750/100 𝑝𝑞−225/100 𝑝^2−625/100 𝑞^2+750/100 𝑝𝑞 = (625/100−225/100) 𝑝^2+(225/100−625/100) 𝑞^2+(−750/100 +750/100)𝑝𝑞 = 400/100 𝑝^2−400/100 𝑞^2 = 𝟒𝒑^𝟐−𝟒𝒒^𝟐 Ex 9.5, 4 Simplify. (vi) (𝑎𝑏+𝑏𝑐)^2−2𝑎𝑏^2 𝑐 (𝑎𝑏+𝑏𝑐)^2−2𝑎𝑏^2 𝑐 (𝑥+𝑦)^2=𝑥^2+𝑦^2+2𝑥𝑦 Putting 𝑥 = 𝑎𝑏 & 𝑦 = 𝑏𝑐 = (𝑎𝑏)^2+(𝑏𝑐)^2+2(𝑎𝑏)(𝑏𝑐)−2𝑎𝑏^2 𝑐 = (𝑎^2×𝑏^2 )+(𝑏^2×𝑐^2 )+(2×𝑎×(𝑏×𝑏)×𝑐)−2𝑎𝑏^2 𝑐 = 𝑎^2 𝑏^2+𝑏^2 𝑐^2+2𝑎𝑏^2 𝑐−2𝑎𝑏^2 𝑐 = 𝒂^𝟐 𝒃^𝟐+𝒃^𝟐 𝒄^𝟐 Ex 9.5, 4 Simplify. (vii) (𝑚^2−𝑛^2 𝑚)^2+2𝑚^3 𝑛^2 (𝑚^2−𝑛^2 𝑚)^2+2𝑚^3 𝑛^2 (𝑎−𝑏)^2=𝑎^2+𝑏^2−2𝑎𝑏 Putting 𝑎 = 𝑚^2 & 𝑏 = 𝑛^2 𝑚 = (𝑚^2 )^2+(𝑛^2 𝑚)^2−2(𝑚^2 )(𝑛^2 𝑚)+2𝑚^3 𝑛^2 = 𝑚^(2 × 2)+(𝑛^2 )^2×𝑚^2−2×(𝑚^2×𝑚)×𝑛^2+2𝑚^3 𝑛^2 = 𝑚^4+𝑛^4 𝑚^2−2𝑚^3 𝑛^2+2𝑚^3 𝑛^2 = 𝒎^𝟒+𝒏^𝟒 𝒎^𝟐