(a) \((2+3\cdot 6)\cdot 5-1\)
(b) \(17-3\cdot(-4-2\cdot(-5))\)
(c) \(5\cdot(3^2-11)^3\)
(d) \(-3^2+(-2)^4\)
(e) \((-5^2+22)^3\)
(f) \(1+(-3)(-2)^2\)
(g) \(1+(-3)(-2^2)\)
(h) \(1+(-3(-2))^2\)
(a) \((-(1-(-2)(-1))-2-(-1)\cdot(-2))\cdot(-1)-2\cdot(-1)\)
(b) \(-\big(-1-(-2^2)\big)^2\cdot(-3)-(-3)^2\big(-2-(-1)\big)^3\)
(c) \(\big(-3-(-3)^2\big)\cdot3-2\cdot\Big(\big(2-(-2)\big)\cdot2-2^2\Big)\)
Solutions: (a) \(5\); (b) \(36\); (c) \(-44\)(a) \(\left(\frac{5}{12}+\frac{7}{8}\right)\cdot\frac{32}{31}\)
(b) \(1\frac{3}{5}\cdot\left(7\frac{1}{8}-7\cdot\frac{1}{8}\right)\)
(c) \(\left(2-\frac{33}{10}\cdot\frac{5}{6}\right)^2\)
(d) \(\left(2\frac{8}{15}-\frac{1}{5}\right)^{-1}\)
(e) \(\left(\frac{13}{10}-\frac{1}{4}\right):\sqrt{2\frac{1}{4}}\)
(f) \(\frac{\textstyle\frac{12}{5}\times\frac{20}{28}}{\textstyle7\frac{1}{5}}\)
(g) \(\frac{\textstyle\frac{5}{12}+1}{\textstyle 3-\frac{1}{6}}\)
(h) \(\frac{\textstyle\frac{2}{3}}{\textstyle ~4~}-\frac{\textstyle ~2~}{\textstyle\frac{3}{4}}\)
(a) \({\displaystyle\frac{2}{1+\frac{1}{5}}\cdot\Big(\frac{4}{75} +\frac{7}{50}\Big)\cdot3-\frac{1}{2}}\)
(b) \({\displaystyle\frac{3-\frac{1}{5}}{3+\frac{1}{5}}+\frac{31}{34}\cdot\frac{17}{62}}\)
(c) \({\displaystyle\left(\frac{1-\frac{1}{4}}{1+\frac{1}{2}} - 2^{-3}\right)\cdot\frac{190}{57} - 1}\)
(d) \(0.22\cdot\big(0.444+5^{-2}\big)^{-1}\)
Solutions: (a) \(\frac{7}{15}\); (b) \(\frac{9}{8}\); (c) \(\frac{1}{4}\); (d) \(\frac{5}{11}\)(a) \((1+\sqrt{5}\,)\sqrt{5}\)
(b) \(\big(1-\sqrt{3}\,\big)^2\)
(c) \((\sqrt{18}-\sqrt{8}\,):15\)
(d) \(\sqrt{20}+\sqrt{45}-\sqrt{5}\)
(e) \(\sqrt{30}\cdot\sqrt{10}-\sqrt{6}\cdot\sqrt{2}\)
(f) \(\sqrt{7+\sqrt{22+\sqrt{9}\,}\,}\)
(g) \({\displaystyle\frac{13}{\sqrt{6}}-\frac{\sqrt{6}}{6}}\)
(h) \({\displaystyle\left(\sqrt{3}-\frac{2}{\sqrt{3}}\right)^3}\)
(a) \((x+2)(x-3)\)
(b) \((x-5)x+3\)
(c) \(1+(x-3)^2\)
(d) \((x-1)(x+2)(x-3)\)
(e) \((2x+1)^2-(x-2)^2\)
(f) \((-x)(x+2)-3(x-5)\)
Solutions: (a) \(x^2-x-6\); (b) \(x^2-5x+3\); (c) \(x^2-6x+10\); (d) \(x^3-2x^2-5x+6\); (e) \(3x^2+8x-3\); (f) \(-x^2-5x+15\)(a) \(a^2(a+1)-(a-1)^2\)
(b) \((a+b)b+a(a-b)\)
(c) \((m+n+1)^2\)
(d) \((p+q)^3\)
Solutions: (a) \(a^3+2a-1\); (b) \(a^2+b^2\); (c) \(m^2+2mn+n^2+2m+2n+1\); (d) \(p^3+3p^2q+3pq^2+q^3\)(a) \(x^2+7x\)
(b) \(5x^4+10x^2\)
(c) \(ab+b^2\)
(d) \(a^3+a^2b-abc\)
Solutions: (a) \(x(x+7)\); (b) \(5x^2(x^2+2)\); (c) \(b(a+b)\); (d) \(a(a^2+ab-bc)\)(a) \(x^2+8x+15\)
(b) \(x^2-8x+12\)
(c) \(a^2+a-6\)
(d) \(a^2-5a+6\)
Solutions: (a) \((x+3)(x+5)\); (b) \((x-6)(x-2)\); (c) \((a+3)(a-2)\); (d) \((a-3)(a-2)\)(a) \(x^3-4x^2-5x\)
(b) \(y^4+2y^3-24y^2\)
(c) \(a^3-6a^2+9a\)
(d) \(4k^3-8k^2-32k\)
Solutions: (a) \(x(x+1)(x-5)\); (b) \(y^2(y-4)(y+6)\); (c) \(a(a-3)^2\); (d) \(4k(k+2)(k-4)\)(a) \(x^2-25\)
(b) \(x^2-9\)
(c) \(4x^2-49\)
(d) \(9m^2-16\)
(e) \(49u^2-v^2\)
Solutions: (a) \((x-5)(x+5)\); (b) \((x-3)(x+3)\); (c) \((2x-7)(2x+7)\); (d) \((3m-4)(3m+4)\); (e) \((7u-v)(7u+v)\)(a) \(x^3-9x\)
(b) \(2a^4-8a^2\)
(c) \(a^3b-ab^3\)
Solutions: (a) \(x(x-3)(x+3)\); (b) \(2a^2(a-2)(a+2)\); (c) \(ab(a-b)(a+b)\)(a) \(x^4-10x^2+9\)
(b) \(x^4+5x^2-36\)
(c) \(a^4-5a^2+4\)
(d) \(a^4+5a^2+4\)
Solutions: (a) \((x-1)(x+1)(x-3)(x+3)\); (b) \((x-2)(x+2)(x^2+9)\); (c) \((a-1)(a+1)(a-2)(a+2)\); (d) \((a^2+1)(a^2+4)\)(a) \(x^3-2x^2-9x+18\)
(b) \(x^3-5x^2+4x-20\)
(c) \(y^3-y^2-25y+25\)
(d) \(y^4-y^3-y^2+y\)
Solutions: (a) \((x-2)(x-3)(x+3)\); (b) \((x-5)(x^2+4)\); (c) \((y-1)(y-5)(y+5)\); (d) \(y(y-1)^2(y+1)\)(a) \(8x-4=5(x+1)\)
(b) \(x^2+2x+8=(x-2)x\)
(c) \((x+2)^2-(x-1)^2= 9\,(x-3)\)
(d) \((x - 1)(x + 2)(x + 3) = x^3+4x^2+8x+16\)
Solutions: (a) \(x=3\); (b) \(x=-2\); (c) \(x=10\); (d) \(x = -\frac{22}{7}\)(a) \({\displaystyle\frac{x}{3}=\frac{x+2}{4}}\)
(b) \({\displaystyle\frac{x}{4}-4x=\frac{3x-7}{2}}\)
(c) \({\displaystyle\frac{x+5}{3} = 1+\frac{2}{3}\,\Big(x+\frac{1}{2}\Big)}\)
(d) \({\displaystyle\frac{1}{4}(x+2)^2-\Big(\frac{x-1}{2}\Big)^2=\frac{1}{4}+4x}\)
(e) \({\displaystyle\Big(\frac{x+1}{2}\Big)^3=\frac{1}{8}(x^3+3x^2)-1}\)
Solutions: (a) \(x=6\); (b) \(x=\frac{2}{3}\); (c) \(x=1\); (d) \(x=\frac{1}{5}\); (e) \(x=-3\)(a) \({\displaystyle\frac{1}{2x+1}=\frac{1}{3x-2}}\)
(b) \({\displaystyle\frac{x+1}{x-1}=\frac{3}{2}}\)
(c) \({\displaystyle\frac{x}{x+2}+\frac{3}{x-2}=1}\)
(d) \({\displaystyle\frac{2x}{x-2}=\frac{x+2}{x-2}}\)
Solutions: (a) \(x=3\); (b) \(x=5\); (c) \(x=-10\); (d) No solution: \(x\) doesn't exist.(a) \((x-2)(x-5)=0\)
(b) \((x-4)(x+1)=0\)
(c) \((x-1)(x-4)(x+6)=0\)
(d) \(x(x-3)(x+7)=0\)
Solutions: (a) \(x_1=2,~ x_2=5\); (b) \(x_1=4,~ x_2=-1\); (c) \(x_1=1,~ x_2=4,~ x_3=-6\); (d) \(x_1=0,~ x_2=3,~ x_3=-7\)(a) \(x^2-5x+6=0\)
(b) \(x^2+2x-15=0\)
(c) \(x^3-2x^2-8x=0\)
Solutions: (a) \(x_1=2,~ x_2=3\); (b) \(x_1=3,~ x_2=-5\); (c) \(x_1=0,~ x_2=-2,~ x_3=4\)(a) \(x^2=x+2\)
(b) \(x^2=6x\)
(c) \(x^3=16x\)
(d) \(x^3+45=5x^2+9x\)
Solutions: (a) \(x_1=2,~ x_2=-1\); (b) \(x_1=0,~ x_2=6\); (c) \(x_1=0,~ x_2=4,~ x_3=-4\); (d) \(x_1=5,~ x_2=3,~ x_3=-3\)(a) \(a^2=b+c,~~~b=?\)
(b) \(\frac{\textstyle e+f}{\textstyle g}=h,~~~ g=?\)
(c) \(\frac{\textstyle m^2}{\textstyle n+p}=r^3,~~~ p=?\)
Solutions: (a) \(b=a^2-c\); (b) \(g=\frac{e+f}{h}\); (c) \(p=\frac{m^2}{r^3}-n\)(a) \(\frac{\textstyle a}{\textstyle b\,c}-d=0,~~~ a=?\)
(b) \(\frac{\textstyle a}{\textstyle b\,c}-d=0,~~~ b=?\)
(c) \(\frac{\textstyle a}{\textstyle b\,c}-d=0,~~~ c=?\)
(d) \(\frac{\textstyle a}{\textstyle b\,c}-d=0,~~~ d=?\)
Solutions: (a) \(a=bcd\); (b) \(b=\frac{a}{cd}\); (c) \(c=\frac{a}{bd}\); (d) \(d=\frac{a}{bc}\)(a) \(v=a\,t,~~~ a=?\)
(b) \(s=\frac{1}{2}\,a\,t^2,~~~ a=?\)
(c) \(\frac{\textstyle 1}{\textstyle a}+\frac{\textstyle 1}{\textstyle b}=\frac{\textstyle 1}{\textstyle f},~~~ a=?\)
Solutions: (a) \(a=\frac{v}{t}\); (b) \(a=\frac{2s}{t^2}\); (c) \(a=\frac{bf}{b-f}\)