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5.7: Nguvu ya utawala wa bidhaa kwa watetezi

  • Page ID
    164578
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    Nguvu ya utawala wa bidhaa kwa watazamaji itashughulika na maneno ambapo bidhaa ya besi hufufuliwa kwa nguvu fulani.

    Ufafanuzi: Nguvu ya Utawala wa Bidhaa kwa Watazamaji

    Kwa idadi yoyote halisi a na b na idadi yoyote n, nguvu ya utawala wa bidhaa kwa exponents ni yafuatayo:

    \((a \cdot b)^n =a^n \cdot b^n\)

    Kurahisisha maneno yafuatayo

    \((a \cdot b)^3\)

    Suluhisho

    \(\begin{aligned} &(a \cdot b) 3&& \text{Given} \\ &a \cdot b \cdot a \cdot b \cdot a \cdot b &&\text{Expand using exponent 3} \\ &a \cdot a \cdot a \cdot b \cdot b \cdot b &&\text{Reorder product using the commutative property.} \\ &a^3b^3 && \text{Simplify to single base.} \end{aligned}\)

    Kurahisisha kujieleza kwa kutumia nguvu ya utawala wa bidhaa kwa exponents.

    \((2x^3y^2 )^2 \)

    Suluhisho

    \(\begin{aligned} &((2x^3y^2 )^2 ) &&\text{Given} \\ &= 2^2 \cdot x^{3\cdot 2 } \cdot y^{2\cdot 2 } &&\text{Power of a product rule applied} \\ &= 4x ^6y^4 &&\text{Simplify by multiplying exponents} \end{aligned}\)

    Kurahisisha kujieleza kwa kutumia nguvu ya utawala wa bidhaa kwa exponents.

    \((2x^5 \cdot 3y)^3\)

    Suluhisho

    \(\begin{aligned} &(2x^5 \cdot 3y)^3 &&\text{Given}\\ &= (6x^5 y)^3 &&\text{If possible, simplify inside the parentheses.} \\ &= 6^3 \cdot x^{5\cdot 3 } \cdot y^3 &&\text{Power of a product rule for exponents applied.} \\ &216x^{15}y^3 &&\text{Simplify by multiplying as needed.} \end{aligned}\)

    Kurahisisha kujieleza kwa kutumia nguvu ya utawala wa bidhaa kwa exponents.

    \((3ab^4 )^{−2}\)

    Suluhisho

    \(\begin{aligned} &(3ab^4 )^{−2 } && \text{Given} \\ &= \dfrac{1 }{(3ab^4)^2 } &&\text{Negative exponent rule applied} \\ &= \dfrac{1 }{3^2 \cdot a^3 \cdot b^{4\cdot 2}}&&\text{Power of a product rule applied.} \\ &\dfrac{1 }{9a^3b^8 } &&\text{Simplify by multiplying as needed.} \end{aligned}\)

    Kurahisisha usemi kwa kutumia nguvu ya utawala wa bidhaa kwa exponents.

    1. \((xy^2 ) ^3\)
    2. \((2xy^2z ^3 ) ^7\)
    3. \((x^ 2 ) ^{−3}\)
    4. \((x ^2y) ^{−3}\)
    5. \((2r ^4 )^ 5\)
    6. \((−2p^ 7 )^ 7\)
    7. \((3k \cdot 2j^2 )^5\)