question_answer 20)

                Find the following squares by using the identities.                 (i) \[{{(b-7)}^{2}}\]                          (ii) \[{{(xy+3z)}^{2}}\]                    (iii) \[{{(6{{x}^{2}}-5y)}^{2}}\]                     (iv) \[{{\left( \frac{2}{3}\,m+\frac{3}{2}n \right)}^{3}}\]                 (v) \[{{(0.4\,p-0.5\,q)}^{2}}\]      (vi) \[{{(2xy\,+5y)}^{2}}\].

Answer:

                (i) \[{{(b-7)}^{2}}\]                          \[=(b-7)\,(b-7)\] \[=b(b-7)\,-7(b-7)\] \[={{b}^{2}}-7b-7b+49\] \[={{b}^{2}}-14b+49\] (ii) \[{{(xy+3z)}^{2}}\]\[=(xy+3z)\,(xy+3z)\] \[=xy\,(xy+3z)+3z(xy+3z)\] \[={{x}^{2}}{{y}^{2}}\,+3xyz\,+3xyz\,+9{{z}^{2}}\] \[={{x}^{2}}{{y}^{2}}\,+6xyz+9{{z}^{2}}\] (iii) \[{{(6{{x}^{2}}-5y)}^{2}}\,=(6{{x}^{2}}-5y)\,(6{{x}^{2}}-5y)\] \[=6{{x}^{2}}\,(6{{x}^{2}}-5y)\,-5y\,(6{{x}^{2}}-5y)\] \[=36{{x}^{4}}\,-30{{x}^{2}}y\,-30{{x}^{2}}y\,+25{{y}^{2}}\] \[=36{{x}^{4}}-60{{x}^{2}}y\,+25{{y}^{2}}\] (iv) \[{{\left( \frac{2}{3}\,m+\frac{3}{2}n \right)}^{3}}\] \[=\left( \frac{2}{3}m+\frac{3}{2}n \right)\,\left( \frac{2}{3}m+\frac{3}{2}n \right)\] \[=\frac{2}{3}\,m\left( \frac{2}{3}m+\frac{3}{2}n \right)+\frac{3}{2}n\,\left( \frac{2}{3}m+\frac{3}{2}n \right)\] \[=\frac{4}{9}{{m}^{4}}+mn+mn+\frac{9}{4}\,{{n}^{2}}\] \[=\frac{4}{9}\,{{m}^{2}}+2\,mn\,+\frac{9}{4}{{n}^{2}}\]                 (v) \[{{(0.4\,p-0.5\,q)}^{2}}\] \[=(0.4\,-0.5q)\,-(0.4p-0.5q)\] \[=0.4p(0.4p-0.5q)-0.5q(0.4p-0.5q)\] \[=0.16{{p}^{2}}\,-0.2pq-0.2pq\,+0.25{{q}^{2}}\] \[=0.16{{p}^{2}}-0.4pq+0.25\,{{q}^{2}}\]                 (vi) \[{{(2xy\,+5y)}^{2}}\] \[=(2xy\,+5y)\,\,(2xy+5y)\] \[=2xy\,(2xy+5y)\,+5y\,(2xy+5y)\] \[=4{{x}^{2}}{{y}^{2}}\,+10x{{y}^{2}}\,+10x{{y}^{2}}+25{{y}^{2}}\] \[=4{{x}^{2}}{{y}^{2}}\,+20x{{y}^{2}}+25{{y}^{2}}\].