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Query

5.9: Watazamaji wa busara

  • Page ID
    164582
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    Watazamaji si mara zote integers. Sehemu hii itaangalia katika kesi ambapo exponent ni idadi ya busara. Wakati exponent ni nambari ya busara, maneno yanaweza kuandikwa kama usemi na radical. Utawala ni kuandika jibu lako kwa fomu sawa na tatizo la awali (ikiwa unapoanza na watazamaji, mwisho na watazamaji, au ukianza na radicals, mwisho na radicals).

    Ufafanuzi: Maonyesho ya busara ya Fomu\(\dfrac{1}{n}\)

    Kwa idadi yoyote halisi\(a\) na idadi yoyote integer\(n\), kujieleza na exponent ya\(\dfrac{1}{n}\) inaweza kuwa walionyesha kama ifuatavyo

    \[a^{\frac{1}{n}} = \sqrt[n]{a} \nonumber \]

    Kumbuka:\(n\) ni index katika radical. \(\sqrt[n]{a}\)inasoma "mzizi wa nth wa

    Kumbuka: Wakati radical haina index inayoonekana, kwa default index ni\(2\) (mizizi mraba). Fahirisi kubwa kuliko\(2\) itakuwa alama juu ya radical.

    1. \((4)^{\frac{1}{2}} = \sqrt{4} = 2\)\(\text{Index is \(2\)kwa default}\)
    2. \( (x)^{\frac{1}{7}} = \sqrt[7]{x}\)\(\text{Index is \(7\)}\)
    3. \((−3y)^{\frac{1}{3}} = \sqrt[3]{(-3y)}\)\(\text{Index is \(3\)}\)

    Sasa, Hebu tuangalie nini kinatokea wakati exponent ni idadi ya busara na nambari\(\neq 1\).

    Ufafanuzi: Maonyesho ya busara ya Fomu\(\dfrac{m}{n}\)

    Kwa idadi yoyote halisi\(a\) na idadi yoyote integer\(n\) na\(m\), kujieleza na exponent ya\(\dfrac{m}{n}\) inaweza kuwa walionyesha kama ifuatavyo

    \[a^{\frac{m}{n}} = \sqrt[n]{a^m} \text{ or } (\sqrt[n]{a})^m \nonumber \]

    Kumbuka:\(n\) ni index katika radical na\(m\) ni nguvu ya msingi.

    Andika zifuatazo kwa fomu kali

    1. \((x)^{\frac{2}{3}} = \sqrt[3]{x^2} = (\sqrt[3]{x})^2\)\(\text{Index is \(3\)na msingi ni alimfufua kwa nguvu ya\(2\).}\)
    2. \((5t)^{\frac{7}{8}} = \sqrt[8]{5t^7} = (\sqrt[8]{5t})^7\)\(\text{Index is \(8\)na msingi ni alimfufua kwa\(7\) nguvu.}\)
    3. \((x)^{\frac{2}{3}} = \sqrt[3]{x^2} = (\sqrt[3]{x})^2\)\(\text{Index is \(3\)na msingi alimfufua kwa nguvu\(2\).}\)
    4. \(\begin{array} &&(z)^{−\frac{5}{9}} &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{Given} \\ &= \dfrac{1}{(z)^{\frac{5}{9}}} &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{Negative exponent rule applied} \\ &= \dfrac{1}{\sqrt[9]{x^5}} \text{ or } \left( \dfrac{1}{\sqrt[9]{x}} \right)^5 &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{Rational exponent written as a radical.} \end{array}\)
    5. \(\left( \dfrac{3}{4} \right)^{\frac{5}{7}} = \sqrt[7]{\dfrac{3}{4}^5}\)\(\text{Rational exponent written as radical with index \(7\)na msingi alimfufua kwa nguvu ya\(5\).}\)

    Andika zifuatazo kwa fomu kali.

    1. \((x)^{\frac{5}{7}}\)
    2. \((xy)^{\frac{9}{8}}\)
    3. \((x)^{\frac{9}{5}}\)
    4. \((z)^{−\frac{11}{13}}\)
    5. \(\left( \dfrac{x}{4} \right)^{\frac{6}{9}}\)
    6. \(6(y)^{\frac{1}{17}}\)
    7. \((6y)^{\frac{1}{17}}\)
    8. \(\left( \dfrac{3}{4} \right)^{\frac{x}{y}}\)
    9. \(\left( \dfrac{7}{4} \right)^{(−\frac{x}{y})}\)