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Query

5.2: Utawala wa Bidhaa kwa Watazamaji

  • Page ID
    164589
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    Ufafanuzi: Utawala wa Bidhaa kwa Watazamaji

    Kwa idadi yoyote halisi\(a\) na idadi chanya\(m\) na\(n\), utawala wa bidhaa kwa exponents ni yafuatayo.

    \(a^m \cdot a^n = a^{m+n}\)

    Kumbuka: Msingi lazima iwe sawa kutumia utawala wa bidhaa.

    Wazo:

    Kutoka sehemu ya mwisho,\(x^3 = \textcolor{blue}{ x \cdot x \cdot x }\qquad x^5 = \textcolor{red}{x \cdot x \cdot x \cdot x \cdot x}\)

    Bidhaa zao

    \(x^3 \cdot x^5 = \textcolor{blue}{x \cdot x \cdot x} \textcolor{red}{\cdot x \cdot x \cdot x \cdot x \cdot x} = x^8\)

    Hivyo,\(x^3 \cdot x^5 = x^{3+5 }= x^8\)

    Tumia utawala wa bidhaa wa watazamaji ili kurahisisha maneno.

    1. \(k^3 \cdot k^9\)
    2. \(\left(\dfrac{2 }{7}\right)^2 \cdot \left(\dfrac{2 }{7}\right)^6\)
    3. \((−2a)^3 \cdot (−2a)^7\)
    4. \(x \cdot x^3 \cdot x^{11}\)
    5. \(y^{13 }\cdot y^{33}\)
    6. \(x^3 \cdot y^2 \cdot x \cdot y^4\)
    Suluhisho
    Ufafanuzi Utawala wa Bidhaa Msingi
    \(k^3 \cdot k^9\) \(k^{3+9}= k^{12}\) \(k\)
    \(\left(\dfrac{2 }{7}\right)^2 \cdot \left(\dfrac{2 }{7}\right)^6\) \(\left( \dfrac{2 }{7}\right)^{2+6 }= \left(\dfrac{2 }{7}\right)^8\) \(\dfrac{2}{7}\)
    \((−2a)^3 \cdot (−2a)^7\) \((−2a)^{3+7 }= (−2a)^{10}\) \(-2a\)
    \(x \cdot x^3 \cdot x^{11}\) \(x ^{1+3+11 }= x^{15}\) \(x\)
    \(y^{13 }\cdot y^{33}\) \(y^{13+33 }= y^46\) \(y\)
    \(x^3 \cdot y^2 \cdot x \cdot y^4\) \(x^{3+1 }\cdot y ^{2+4 }= x^{ 4 }\cdot y^{6}\) \(x\)na\(y\)

    Kumbuka: Tena, besi lazima iwe sawa ili kurahisisha kutumia utawala wa bidhaa wa exponent

    Hatua muhimu ili kurahisisha kutumia utawala wa bidhaa wa exponents:

    1. Tambua maneno na misingi ya kawaida
    2. Kutambua exponent ya besi ya kawaida.
    3. Kuongeza exponents ya besi ya kawaida na kufanya matokeo ya jumla exponent mpya.
    4. Kurudia hatua kama unahitaji

    Tumia utawala wa bidhaa wa wafuatiliaji ili kurahisisha zifuatazo.

    1. \(f^3 \cdot f^11\)
    2. \(\left(\dfrac{x}{7}\right)^2 \cdot \left(\dfrac{x }{7}\right)^3\)
    3. \((−7x)^9 \cdot (−7x)^7\)
    4. \(h^5 \cdot h^3 \cdot h^{11}\)
    5. \(t^{13} \cdot t^{33}\)
    6. \(x^8 \cdot y^2 \cdot z \cdot x^ 3 \cdot y^2 \cdot z^{17}\)
    7. \(x^3 \cdot y^4 \cdot x^3\)