Numbers, expressions and equations

Integers, fractions and real numbers

1. First calculate the following calculations without a calculator – “manually”.
   Then verify your answers with your GDC (Graphic Display Calculator).
   

   (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)   $1+(-3(-2){)}^2$

   Solutions:    (a)  $99$;     (b)  $-1$;     (c)  $-40$;     (d)  $7$;     (e)  $-27$;     (f)  $-11$;     (g)  $13$;     (h)  $37$
2. First calculate the following calculations without a calculator – “manually”.
   Then verify your answers with your GDC.
   

   (a)   $(-(1-(-2)(-1))-2-(-1)\cdot (-2))\cdot (-1)-2\cdot (-1)$

   (b)   $-(-1-(-2^2){)}^2\cdot (-3)-(-3{)}^2(-2-(-1){)}^3$

   (c)   $(-3-(-3{)}^2)\cdot 3-2\cdot ((2-(-2))\cdot 2-2^2)$

   Solutions:    (a)  $5$;     (b)  $36$;     (c)  $-44$
3. First calculate the following calculations without a calculator and write the result as a fraction (or as a mixed number or an integer).
   Then verify your answers with your GDC.
   

   (a)   $(\frac{5}{12}+\frac{7}{8})\cdot \frac{32}{31}$

   (b)   $1\frac{3}{5}\cdot (7\frac{1}{8}-7\cdot \frac{1}{8})$

   (c)   ${(2-\frac{33}{10}\cdot \frac{5}{6})}^2$

   (d)   ${(2\frac{8}{15}-\frac{1}{5})}^{-1}$

   (e)   $(\frac{13}{10}-\frac{1}{4}):\sqrt{2\frac{1}{4}}$

   (f)   $\frac{\frac{12}{5}\times \frac{20}{28}}{7\frac{1}{5}}$

   (g)   $\frac{\frac{5}{12}+1}{3-\frac{1}{6}}$

   (h)   $\frac{\frac{2}{3}}{4}-\frac{2}{\frac{3}{4}}$

   Solutions:    (a)  $\frac{4}{3}$  or  $1\frac{1}{3}$;     (b)  $10$;     (c)  $\frac{9}{16}$;     (d)  $\frac{3}{7}$;     (e)  $\frac{7}{10}$;     (f)  $\frac{5}{21}$;     (g)  $\frac{1}{2}$;     (h)  $-\frac{5}{2}$  or  $-2\frac{1}{2}$
4. First calculate the following calculations without a calculator and write the result as a reduced fraction.
   Then verify your answers with your GDC.
   

   (a)   ${\displaystyle \frac{2}{1+\frac{1}{5}}\cdot (\frac{4}{75}+\frac{7}{50})\cdot 3-\frac{1}{2}}$

   (b)   ${\displaystyle \frac{3-\frac{1}{5}}{3+\frac{1}{5}}+\frac{31}{34}\cdot \frac{17}{62}}$

   (c)   ${\displaystyle (\frac{1-\frac{1}{4}}{1+\frac{1}{2}}-2^{-3})\cdot \frac{190}{57}-1}$

   (d)   $0.22\cdot (0.444+5^{-2}{)}^{-1}$

   Solutions:    (a)  $\frac{7}{15}$;     (b)  $\frac{9}{8}$;     (c)  $\frac{1}{4}$;     (d)  $\frac{5}{11}$
5. First calculate the following calculations without a calculator and write the result in exact form (using fractions and roots).
   Then verify your answers with your GDC and write the results rounded to three significant figures.
   

   (a)   $(1+\sqrt{5})\sqrt{5}$

   (b)   $(1-\sqrt{3}{)}^2$

   (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)   ${\displaystyle {(\sqrt{3}-\frac{2}{\sqrt{3}})}^3}$

   Solutions:    (a)  $5+\sqrt{5}\approx 7.24$;     (b)  $4-2\sqrt{3}\approx 0.536$;     (c)  $\frac{\sqrt{2}}{15}\approx 0.0943$;     (d)  $4\sqrt{5}\approx 8.94$;     (e)  $8\sqrt{3}\approx 13.9$;     (f)  $2\sqrt{3}\approx 3.46$;     (g)  $2\sqrt{6}\approx 4.90$;     (h)  $\frac{\sqrt{3}}{9}\approx 0.192$

Algebraic expressions

6. Expand the following expressions (remove the brackets and write each expression as a sum of terms):
   

   (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)  $-x^2-5x+15$
7. Expand the following expressions:
   

   (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)  $p^3+3p^{2}q+3p{q}^2+q^3$
8. Factorise the following expressions (write each expression as a product of factors).
   Hint: factor out the common factor.
   

   (a)   $x^2+7x$

   (b)   $5x^4+10x^2$

   (c)   $ab+b^2$

   (d)   $a^3+a^{2}b-abc$

   Solutions:    (a)  $x(x+7)$;     (b)  $5x^2(x^2+2)$;     (c)  $b(a+b)$;     (d)  $a(a^2+ab-bc)$
9. Factorise the following expressions.
   Hint: use the Vieta's rule.
   

   (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)  $(a-3)(a-2)$
10. Factorise the following expressions:
    

    (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)  $4k(k+2)(k-4)$
11. Factorise the following expressions:
    Hint: use the formula $a^2-b^2=(a-b)(a+b)$.
    

    (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)  $(7u-v)(7u+v)$
12. Factorise the following expressions:
    

    (a)   $x^3-9x$

    (b)   $2a^4-8a^2$

    (c)   $a^{3}b-a{b}^3$

    Solutions:    (a)  $x(x-3)(x+3)$;     (b)  $2a^2(a-2)(a+2)$;     (c)  $ab(a-b)(a+b)$
13. Factorise the following expressions:
    

    (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)  $(a^2+1)(a^2+4)$
14. Factorise the following expressions:
    

    (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)  $y(y-1{)}^2(y+1)$

Equations

15. Solve the following equations:
    

    (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)  $x=-\frac{22}{7}$
16. Solve the following equations:
    

    (a)   ${\displaystyle \frac{x}{3}=\frac{x+2}{4}}$

    (b)   ${\displaystyle \frac{x}{4}-4x=\frac{3x-7}{2}}$

    (c)   ${\displaystyle \frac{x+5}{3}=1+\frac{2}{3}(x+\frac{1}{2})}$

    (d)   ${\displaystyle \frac{1}{4}(x+2{)}^2-(\frac{x-1}{2}{)}^2=\frac{1}{4}+4x}$

    (e)   ${\displaystyle (\frac{x+1}{2}{)}^3=\frac{1}{8}(x^3+3x^2)-1}$

    Solutions:    (a)  $x=6$;     (b)  $x=\frac{2}{3}$;     (c)  $x=1$;     (d)  $x=\frac{1}{5}$;     (e)  $x=-3$
17. Solve the following equations:
    

    (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.
18. Solve the following equations.
    Hint: Use zero product property (when product is equal to zero – what can you tell about factors?).
    

    (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)  $x_1=0,x_2=3,x_3=-7$
19. Solve the following equations using factorisation:
    

    (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)  $x_1=0,x_2=-2,x_3=4$
20. Solve the following equations:
    

    (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)  $x_1=5,x_2=3,x_3=-3$
21. Express the chosen variable out of the given equation:
    

    (a)   $a^2=b+c,b=?$

    (b)   $\frac{e+f}{g}=h,g=?$

    (c)   $\frac{m^2}{n+p}=r^3,p=?$

    Solutions:    (a)  $b=a^2-c$;     (b)  $g=\frac{e+f}{h}$;     (c)  $p=\frac{m^2}{r^3}-n$
22. Express the chosen variable out of the given equation:
    

    (a)   $\frac{a}{bc}-d=0,a=?$

    (b)   $\frac{a}{bc}-d=0,b=?$

    (c)   $\frac{a}{bc}-d=0,c=?$

    (d)   $\frac{a}{bc}-d=0,d=?$

    Solutions:    (a)  $a=bcd$;     (b)  $b=\frac{a}{cd}$;     (c)  $c=\frac{a}{bd}$;     (d)  $d=\frac{a}{bc}$
23. Solve given equations for variable $a$:
    

    (a)   $v=at,a=?$

    (b)   $s=\frac{1}{2}a{t}^2,a=?$

    (c)   $\frac{1}{a}+\frac{1}{b}=\frac{1}{f},a=?$

    Solutions:    (a)  $a=\frac{v}{t}$;     (b)  $a=\frac{2s}{t^2}$;     (c)  $a=\frac{bf}{b-f}$

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