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5.8: Nguvu ya utawala wa quotient kwa wafuasi

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    164586

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    Nguvu ya utawala wa quotient kwa watazingatia kile kinachotokea kwa quotient wakati inafufuliwa kwa nguvu fulani.

    Ufafanuzi: Nguvu ya Utawala wa Quotient kwa Watazamaji

    Kwa idadi yoyote halisi\(a\) na\(b\) na integer yoyote\(n\), nguvu ya utawala quotient kwa exponents ni yafuatayo:

    \(\left( \dfrac{a }{b} \right)^n = \dfrac{a^n }{b^n }\),

    wapi\(b \neq 0\).

    Kurahisisha zifuatazo kwa kutumia nguvu ya utawala quotient kwa exponents.

    Mfano Template:index

    Kurahisisha zifuatazo kwa kutumia nguvu ya utawala quotient kwa exponents.

    \(\left( \dfrac{a }{b} \right)^4\)

    Suluhisho

    \(\begin{aligned} &\left( \dfrac{a}{ b} \right)^4 && \text{Given} \\ &= \dfrac{a }{b} \cdot \dfrac{a }{b} \cdot \dfrac{a }{b} \cdot \dfrac{a }{b} &&\text{Expand using the exponent definition} \\ &= \dfrac{a^4 }{b^4} && \text{Multiply as needed to simplify} \end{aligned}\)

    Mfano Template:index

    \(\left(\dfrac{x^2 }{3y^5} \right)^3\)

    Suluhisho

    \(\begin{aligned} &\left( \dfrac{x^2 }{3y^5 }\right)^3 && \text{Given} \\ &= \dfrac{x^{2\cdot 3 }}{3^3 \cdot y^{5\cdot 3 }} && \text{power of quotient rule for exponents applied} \\ &= \dfrac{x^6 }{3^3 \cdot y^{15 }} &&\text{Simplify exponent product} \\ &= \dfrac{x^6 }{27y^{15 }} && \text{Multiply as needed to simplify.} \end{aligned}\)

    Mfano Template:index

    \(\left( \dfrac{2x }{y }\right)^{−3}\)

    Suluhisho

    \(\begin{aligned} &\left( \dfrac{2x }{y }\right)^{−3 } &&\text{Given} \\ &= \left( \dfrac{y }{2x} \right)^3 && \text{Negative exponent rule applied} \\ &= \dfrac{y^3 }{2^3 \cdot x^3} && \text{Power of a quotient rule for exponents applied.} \\ &= \dfrac{y^3 }{8x^3 } && \text{Multiply as needed to simplify.} \end{aligned}\)

    Utaratibu ambao sheria za watazamaji hutumiwa haijalishi. Kwa mfano tatu, hatua 2 na 3 zinaweza kufanywa kwa utaratibu wowote. Matokeo yatakuwa sawa.

    Zoezi Template:index

    Kurahisisha usemi kwa kutumia nguvu ya utawala quotient kwa exponents.

    1. \(\left( \dfrac{p^4 }{p^7 }\right) ^3\)
    2. \(−\left(\dfrac{ x^2 \cdot x^3 }{x \cdot y^3} \right) ^2\)
    3. \(\left( \dfrac{5x^3 }{2y^{13 }}\right) ^{−2}\)
    4. \(\left( \dfrac{2c^3}{ c^4} \right) ^3\)
    5. \(\left( \dfrac{a ^{−7}b }{a^2b^{−4 }}\right)^3\)
    6. \(\left( \dfrac{f^{−7 }}{f^5 }\right)^9\)
    7. \(\left(\dfrac{ xy^2z^3}{ x^3y^2z} \right) ^5\)