Numbers and expressions

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)   \(-\big(-1-(-2^2)\big)^2\cdot(-3)-(-3)^2\big(-2-(-1)\big)^3\)

   (c)   \(\big(-3-(-3)^2\big)\cdot3-2\cdot\Big(\big(2-(-2)\big)\cdot2-2^2\Big)\)

   Solutions:    (a)  \(5\);     (b)  \(36\);     (c)  \(-44\)
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)   \(\left(\frac{5}{12}+\frac{7}{8}\right)\cdot\frac{32}{31}\)

   (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)   \(\frac{\textstyle\frac{2}{3}}{\textstyle ~4~}-\frac{\textstyle ~2~}{\textstyle\frac{3}{4}}\)

   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)   \({\displaystyle\frac{2}{1+\frac{1}{5}}\cdot\Big(\frac{4}{75} +\frac{7}{50}\Big)\cdot3-\frac{1}{2}}\)

   (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)  \(\frac{5}{11}\)
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)   \(\big(1-\sqrt{3}\,\big)^2\)

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

   Solutions:    (a)  \(5+\sqrt{5}\approx7.24\);     (b)  \(4-2\sqrt{3}\approx0.536\);     (c)  \(\frac{\sqrt{2}}{15}\approx0.0943\);     (d)  \(4\sqrt{5}\approx8.94\);     (e)  \(8\sqrt{3}\approx13.9\);     (f)  \(2\sqrt{3}\approx3.46\);     (g)  \(2\sqrt{6}\approx4.90\);     (h)  \(\frac{\sqrt{3}}{9}\approx0.192\)
6. Calculate the following calculations using your GDC and write the results rounded to three significant figures.
   

   (a)   \({\displaystyle\frac{7}{\sqrt{6}+\sqrt{5}}\cdot\frac{\frac{17}{16}}{~15~}}\)

   (b)   \({\displaystyle\left(\frac{2^3+3^2}{\sqrt{3}-\sqrt{2}}\right)^{-2}}\)

   (c)   \({\displaystyle\frac{12^4}{(\sqrt{10}-3)^3}}\)

   Solutions:    (a)  \(0.106\);     (b)  \(0.000350\)  or  \(3.50\cdot10^{-4}\);     (c)  \(4.85\cdot10^6\)
7. Given the numbers \(a=0.0022349\),  \(b=\sqrt{\frac{12}{3\,456\,789}}\) and \(c=1.5867\cdot 10^7\) calculate the following calculations using your GDC and write the results rounded to three significant figures.
   

   (a)   \(abc\)

   (b)   \(abc^{-1}\)

   (c)   \((abc)^{-1}\)

   (d)   \((a-b)c\)

   Hint:    First store the given numbers into your calculator's memory.
   Solutions:    (a)  \(abc\approx 66.1\);     (b)  \(abc^{-1}\approx2.62\cdot 10^{-13}\);     (c)  \((abc)^{-1}\approx0.0151\) or \(1.51\cdot 10^{-2}\);     (d)  \((a-b)c\approx5.90\cdot10^3\)

Algebraic expressions

8. Consider the expression  \((x+1)^2-7x+3\).
   Substitute the following values for \(x\) and evaluate the expression:
   

   (a)   \(x=1\)

   (b)   \(x=3\)

   (c)   \(x=-2\)

   (d)   \(x=\frac{1}{3}\)

   Solutions:    (a)  \(0\);     (b)  \(-2\);     (c)  \(18\);     (d)  \(\frac{22}{9}\)
9. Consider the expression  \(x^4+x^3-7x^2+6x+10\).
   Substitute the following values for \(x\) and evaluate the expression. Write the results in exact form.
   

   (a)   \(x=\sqrt{2}\)

   (b)   \(x=\sqrt{5}\)

   (c)   \(x=\sqrt{3}\)

   Solutions:    (a)  \(8\sqrt{2}\);     (b)  \(11\sqrt{5}\);     (c)  \(-2+9\sqrt{3}\)
10. 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)  \(-x^2-5x+15\)
11. 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)  \(p^3+3p^2q+3pq^2+q^3\)
12. 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)   \(a^3+a^2b-abc\)

    Solutions:    (a)  \(x(x+7)\);     (b)  \(5x^2(x^2+2)\);     (c)  \(b(a+b)\);     (d)  \(a(a^2+ab-bc)\)
13. 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)  \((a-3)(a-2)\)
14. 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)  \(4k(k+2)(k-4)\)
15. Factorise the following expressions:
    

    (a)   \(2x^2-11x+15\)

    (b)   \(3x^2-8x+4\)

    (c)   \(3x^3+2x^2-x\)

    Solutions:    (a)  \((2x-5)(x-3)\);     (b)  \((3x-2)(x-2)\);     (c)  \(x(3x-1)(x+1)\)
16. 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)  \((7u-v)(7u+v)\)
17. Factorise the following expressions:
    

    (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)  \(ab(a-b)(a+b)\)
18. Factorise the following expressions:
    

    (a)   \((x+1)^2-y^2\)

    (b)   \(4x^2-(x-7)^2\)

    (c)   \((2x+5)^2-(x-1)^2\)

    Solutions:    (a)  \((x-y+1)(x+y+1)\);     (b)  \((x+7)(3x-7)\);     (c)  \((3x+4)(x+6)\)
19. 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)  \((a^2+1)(a^2+4)\)
20. 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)  \(y(y-1)^2(y+1)\)

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