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8.3E: Mazoezi

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    Mazoezi hufanya kamili

    Zoezi SET A: tumia mali ya bidhaa ili kurahisisha maneno makubwa

    Katika mazoezi yafuatayo, tumia Mali ya Bidhaa ili kurahisisha maneno makubwa.

    1. \(\sqrt{27}\)
    2. \(\sqrt{80}\)
    3. \(\sqrt{125}\)
    4. \(\sqrt{96}\)
    5. \(\sqrt{147}\)
    6. \(\sqrt{450}\)
    7. \(\sqrt{800}\)
    8. \(\sqrt{675}\)
      1. \(\sqrt[4]{32}\)
      2. \(\sqrt[5]{64}\)
      1. \(\sqrt[3]{625}\)
      2. \(\sqrt[6]{128}\)
      1. \(\sqrt[5]{64}\)
      2. \(\sqrt[3]{256}\)
      1. \(\sqrt[4]{3125}\)
      2. \(\sqrt[3]{81}\)
    Jibu

    1. \(3\sqrt{3}\)

    3. \(5\sqrt{5}\)

    5. \(7\sqrt{3}\)

    7. \(20\sqrt{2}\)

    9.

    1. \(2 \sqrt[4]{2}\)
    2. \(2 \sqrt[5]{2}\)

    11.

    1. \(2 \sqrt[5]{2}\)
    2. \(4 \sqrt[3]{4}\)
    Zoezi SET B: tumia mali ya bidhaa ili kurahisisha maneno makubwa

    Katika mazoezi yafuatayo, kurahisisha kutumia ishara za thamani kamili kama inahitajika.

      1. \(\sqrt{y^{11}}\)
      2. \(\sqrt[3]{r^{5}}\)
      3. \(\sqrt[4]{s^{10}}\)
      1. \(\sqrt{m^{13}}\)
      2. \(\sqrt[5]{u^{7}}\)
      3. \(\sqrt[6]{v^{11}}\)
      1. \(\sqrt{n^{21}}\)
      2. \(\sqrt[3]{q^{8}}\)
      3. \(\sqrt[8]{n^{10}}\)
      1. \(\sqrt{r^{25}}\)
      2. \(\sqrt[5]{p^{8}}\)
      3. \(\sqrt[4]{m^{5}}\)
      1. \(\sqrt{125 r^{13}}\)
      2. \(\sqrt[3]{108 x^{5}}\)
      3. \(\sqrt[4]{48 y^{6}}\)
      1. \(\sqrt{80 s^{15}}\)
      2. \(\sqrt[5]{96 a^{7}}\)
      3. \(\sqrt[6]{128 b^{7}}\)
      1. \(\sqrt{242 m^{23}}\)
      2. \(\sqrt[4]{405 m 10}\)
      3. \(\sqrt[5]{160 n^{8}}\)
      1. \(\sqrt{175 n^{13}}\)
      2. \(\sqrt[5]{512 p^{5}}\)
      3. \(\sqrt[4]{324 q^{7}}\)
      1. \(\sqrt{147 m^{7} n^{11}}\)
      2. \(\sqrt[3]{48 x^{6} y^{7}}\)
      3. \(\sqrt[4]{32 x^{5} y^{4}}\)
      1. \(\sqrt{96 r^{3} s^{3}}\)
      2. \(\sqrt[3]{80 x^{7} y^{6}}\)
      3. \(\sqrt[4]{80 x^{8} y^{9}}\)
      1. \(\sqrt{192 q^{3} r^{7}}\)
      2. \(\sqrt[3]{54 m^{9} n^{10}}\)
      3. \(\sqrt[4]{81 a^{9} b^{8}}\)
      1. \(\sqrt{150 m^{9} n^{3}}\)
      2. \(\sqrt[3]{81 p^{7} q^{8}}\)
      3. \(\sqrt[4]{162 c^{11} d^{12}}\)
      1. \(\sqrt[3]{-864}\)
      2. \(\sqrt[4]{-256}\)
      1. \(\sqrt[5]{-486}\)
      2. \(\sqrt[6]{-64}\)
      1. \(\sqrt[5]{-32}\)
      2. \(\sqrt[8]{-1}\)
      1. \(\sqrt[3]{-8}\)
      2. \(\sqrt[4]{-16}\)
      1. \(5+\sqrt{12}\)
      2. \(\dfrac{10-\sqrt{24}}{2}\)
      1. \(8+\sqrt{96}\)
      2. \(\dfrac{8-\sqrt{80}}{4}\)
      1. \(1+\sqrt{45}\)
      2. \(\dfrac{3+\sqrt{90}}{3}\)
      1. \(3+\sqrt{125}\)
      2. \(\dfrac{15+\sqrt{75}}{5}\)
    Jibu

    1.

    1. \(\left|y^{5}\right| \sqrt{y}\)
    2. \(r \sqrt[3]{r^{2}}\)
    3. \(s^{2} \sqrt[4]{s^{2}}\)

    3.

    1. \(n^{10} \sqrt{n}\)
    2. \(q^{2} \sqrt[3]{q^{2}}\)
    3. \(|n| \sqrt[8]{n^{2}}\)

    5.

    1. \(5 r^{6} \sqrt{5 r}\)
    2. \(3 x \sqrt[3]{4 x^{2}}\)
    3. \(2|y| \sqrt[4]{3 y^{2}}\)

    7.

    1. \(11\left|m^{11}\right| \sqrt{2 m}\)
    2. \(3 m^{2} \sqrt[4]{5 m^{2}}\)
    3. \(2 n \sqrt[5]{5 n^{3}}\)

    9.

    1. \(7\left|m^{3} n^{5}\right| \sqrt{3 m n}\)
    2. \(2 x^{2} y^{2} \sqrt[3]{6 y}\)
    3. \(2|x y| \sqrt[4]{2 x}\)

    11.

    1. \(8\left|q r^{3}\right| \sqrt{3 q r}\)
    2. \(3 m^{3} n^{3} \sqrt[3]{2 n}\)
    3. \(3 a^{2} b^{2} \sqrt[4]{a}\)

    13.

    1. \(-6 \sqrt[3]{4}\)
    2. si kweli

    15.

    1. \(-2\)
    2. si kweli

    17.

    1. \(5+2 \sqrt{3}\)
    2. \(5-\sqrt{6}\)

    19.

    1. \(1+3 \sqrt{5}\)
    2. \(1+\sqrt{10}\)
    Zoezi Kuweka C: tumia mali ya quotient ili kurahisisha maneno makubwa

    Katika mazoezi yafuatayo, tumia Mali ya Quotient ili kurahisisha mizizi ya mraba.

      1. \(\sqrt{\dfrac{45}{80}}\)
      2. \(\sqrt[3]{\dfrac{8}{27}}\)
      3. \(\sqrt[4]{\dfrac{1}{81}}\)
      1. \(\sqrt{\dfrac{72}{98}}\)
      2. \(\sqrt[3]{\dfrac{24}{81}}\)
      3. \(\sqrt[4]{\dfrac{6}{96}}\)
      1. \(\sqrt{\dfrac{100}{36}}\)
      2. \(\sqrt[3]{\dfrac{81}{375}}\)
      3. \(\sqrt[4]{\dfrac{1}{256}}\)
      1. \(\sqrt{\dfrac{121}{16}}\)
      2. \(\sqrt[3]{\dfrac{16}{250}}\)
      3. \(\sqrt[4]{\dfrac{32}{162}}\)
      1. \(\sqrt{\dfrac{x^{10}}{x^{6}}}\)
      2. \(\sqrt[3]{\dfrac{p^{11}}{p^{2}}}\)
      3. \(\sqrt[4]{\dfrac{q^{17}}{q^{13}}}\)
      1. \(\sqrt{\dfrac{p^{20}}{p^{10}}}\)
      2. \(\sqrt[5]{\dfrac{d^{12}}{d^{7}}}\)
      3. \(\sqrt[8]{\dfrac{m^{12}}{m^{4}}}\)
      1. \(\sqrt{\dfrac{y^{4}}{y^{8}}}\)
      2. \(\sqrt[5]{\dfrac{u^{21}}{u^{11}}}\)
      3. \(\sqrt[6]{\dfrac{v^{30}}{v^{12}}}\)
      1. \(\sqrt{\dfrac{q^{8}}{q^{14}}}\)
      2. \(\sqrt[3]{\dfrac{r^{14}}{r^{5}}}\)
      3. \(\sqrt[4]{\dfrac{c^{21}}{c^{9}}}\)
    9. \(\sqrt{\dfrac{96 x^{7}}{121}}\)
    10. \(\sqrt{\dfrac{108 y^{4}}{49}}\)
    11. \(\sqrt{\dfrac{300 m^{5}}{64}}\)
    12. \(\sqrt{\dfrac{125 n^{7}}{169}}\)
    13. \(\sqrt{\dfrac{98 r^{5}}{100}}\)
    14. \(\sqrt{\dfrac{180 s^{10}}{144}}\)
    15. \(\sqrt{\dfrac{28 q^{6}}{225}}\)
    16. \(\sqrt{\dfrac{150 r^{3}}{256}}\)
      1. \(\sqrt{\dfrac{75 r^{9}}{s^{8}}}\)
      2. \(\sqrt[3]{\dfrac{54 a^{8}}{b^{3}}}\)
      3. \(\sqrt[4]{\dfrac{64 c^{5}}{d^{4}}}\)
      1. \(\sqrt{\dfrac{72 x^{5}}{y^{6}}}\)
      2. \(\sqrt[5]{\dfrac{96 r^{11}}{s^{5}}}\)
      3. \(\sqrt[6]{\dfrac{128 u^{7}}{v^{12}}}\)
      1. \(\sqrt{\dfrac{28 p^{7}}{q^{2}}}\)
      2. \(\sqrt[3]{\dfrac{81 s^{8}}{t^{3}}}\)
      3. \(\sqrt[4]{\dfrac{64 p^{15}}{q^{12}}}\)
      1. \(\sqrt{\dfrac{45 r^{3}}{s^{10}}}\)
      2. \(\sqrt[3]{\dfrac{625 u^{10}}{v^{3}}}\)
      3. \(\sqrt[4]{\dfrac{729 c^{21}}{d^{8}}}\)
      1. \(\sqrt{\dfrac{32 x^{5} y^{3}}{18 x^{3} y}}\)
      2. \(\sqrt[3]{\dfrac{5 x^{6} y^{9}}{40 x^{5} y^{3}}}\)
      3. \(\sqrt[4]{\dfrac{5 a^{8} b^{6}}{80 a^{3} b^{2}}}\)
      1. \(\sqrt{\dfrac{75 r^{6} s^{8}}{48 r s^{4}}}\)
      2. \(\sqrt[3]{\dfrac{24 x^{8} y^{4}}{81 x^{2} y}}\)
      3. \(\sqrt[4]{\dfrac{32 m^{9} n^{2}}{162 m n^{2}}}\)
      1. \(\sqrt{\dfrac{27 p^{2} q}{108 p^{4} q^{3}}}\)
      2. \(\sqrt[3]{\dfrac{16 c^{5} d^{7}}{250 c^{2} d^{2}}}\)
      3. \(\sqrt[6]{\dfrac{2 m^{9} n^{7}}{128 m^{3} n}}\)
      1. \(\sqrt{\dfrac{50 r^{5} s^{2}}{128 r^{2} s^{6}}}\)
      2. \(\sqrt[3]{\dfrac{24 m^{9} n^{7}}{375 m^{4} n}}\)
      3. \(\sqrt[4]{\dfrac{81 m^{2} n^{8}}{256 m^{1} n^{2}}}\)
      1. \(\dfrac{\sqrt{45 p^{9}}}{\sqrt{5 q^{2}}}\)
      2. \(\dfrac{\sqrt[4]{64}}{\sqrt[4]{2}}\)
      3. \(\dfrac{\sqrt[5]{128 x^{8}}}{\sqrt[5]{2 x^{2}}}\)
      1. \(\dfrac{\sqrt{80 q^{5}}}{\sqrt{5 q}}\)
      2. \(\dfrac{\sqrt[3]{-625}}{\sqrt[3]{5}}\)
      3. \(\dfrac{\sqrt[4]{80 m^{7}}}{\sqrt[4]{5 m}}\)
      1. \(\dfrac{\sqrt{50 m^{7}}}{\sqrt{2 m}}\)
      2. \(\sqrt[3]{\dfrac{1250}{2}}\)
      3. \(\sqrt[4]{\dfrac{486 y^{9}}{2 y^{3}}}\)
      1. \(\dfrac{\sqrt{72 n^{11}}}{\sqrt{2 n}}\)
      2. \(\sqrt[3]{\dfrac{162}{6}}\)
      3. \(\sqrt[4]{\dfrac{160 r^{10}}{5 r^{3}}}\)
    Jibu

    1.

    1. \(\dfrac{3}{4}\)
    2. \(\dfrac{2}{3}\)
    3. \(\dfrac{1}{3}\)

    3.

    1. \(\dfrac{5}{3}\)
    2. \(\dfrac{3}{5}\)
    3. \(\dfrac{1}{4}\)

    5.

    1. \(x^{2}\)
    2. \(p^{3}\)
    3. \(|q|\)

    7.

    1. \(\dfrac{1}{y^{2}}\)
    2. \(u^{2}\)
    3. \(|v^{3}|\)

    9. \(\dfrac{4\left|x^{3}\right| \sqrt{6 x}}{11}\)

    11. \(\dfrac{10 m^{2} \sqrt{3 m}}{8}\)

    13. \(\dfrac{7 r^{2} \sqrt{2 r}}{10}\)

    15. \(\dfrac{2\left|q^{3}\right| \sqrt{7}}{15}\)

    17.

    1. \(\dfrac{5 r^{4} \sqrt{3 r}}{s^{4}}\)
    2. \(\dfrac{3 a^{2} \sqrt[3]{2 a^{2}}}{|b|}\)
    3. \(\dfrac{2|c| \sqrt[4]{4 c}}{|d|}\)

    19.

    1. \(\dfrac{2\left|p^{3}\right| \sqrt{7 p}}{|q|}\)
    2. \(\dfrac{3 s^{2} \sqrt[3]{3 s^{2}}}{t}\)
    3. \(\dfrac{2\left|p^{3}\right| \sqrt[4]{4 p^{3}}}{\left|q^{3}\right|}\)

    21.

    1. \(\dfrac{4|x y|}{3}\)
    2. \(\dfrac{y^{2} \sqrt[3]{x}}{2}\)
    3. \(\dfrac{|a b| \sqrt[4]{a}}{4}\)

    23.

    1. \(\dfrac{1}{2|p q|}\)
    2. \(\dfrac{2 c d \sqrt[5]{2 d^{2}}}{5}\)
    3. \(\dfrac{|m n| \sqrt[6]{2}}{2}\)

    25.

    1. \(\dfrac{3 p^{4} \sqrt{p}}{|q|}\)
    2. \(2 \sqrt[4]{2}\)
    3. \(2 x \sqrt[5]{2 x}\)

    27.

    1. \(5\left|m^{3}\right|\)
    2. \(5 \sqrt[3]{5}\)
    3. \(3|y| \sqrt[4]{3 y^{2}}\)
    Zoezi SET D: mazoezi ya kuandika
    1. Eleza kwa nini\(\sqrt{x^{4}}=x^{2}\). Kisha kueleza kwa nini\(\sqrt{x^{16}}=x^{8}\).
    2. Eleza kwa nini\(7+\sqrt{9}\) si sawa na\(\sqrt{7+9}\).
    3. Eleza jinsi unavyojua hilo\(\sqrt[5]{x^{10}}=x^{2}\).
    4. Eleza\(\sqrt[4]{-64}\) kwa nini si idadi halisi lakini\(\sqrt[3]{-64}\) ni.
    Jibu

    1. Majibu inaweza kutofautiana

    3. Majibu inaweza kutofautiana

    Self Check

    Baada ya kukamilisha mazoezi, tumia orodha hii ili kutathmini ujuzi wako wa malengo ya sehemu hii.

    Jedwali hili lina safu 3 na nguzo 4. Mstari wa kwanza ni mstari wa kichwa na huandika kila safu. Kichwa cha kwanza cha safu ni â € naweza â € €, pili ni â € € confidentlyâ €, ya tatu ni â € € na baadhi ya msaada â € €, na ya nne ni â € no, mimi donâ €™ t kupata hiyo €. Chini ya safu ya kwanza ni maneno †kutumia mali ya bidhaa ili kurahisisha expressions radical â € na â € kutumia mali quotient kurahisisha expressions radical â €. Nguzo nyingine zimeachwa tupu ili mwanafunzi aweze kuonyesha kiwango chao cha ustadi kwa kila mada.
    Kielelezo 8.2.1

    b Baada ya kuchunguza orodha hii, utafanya nini ili uwe na ujasiri kwa malengo yote?