Skip to main content
Query

5.1: Ufafanuzi wa a

  • Page ID
    164577
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    Ufafanuzi:\(a^n\)

    Kwa idadi yoyote halisi\(a\) na idadi nzuri\(n\),\(a^n\) ni kuzidisha mara kwa mara ya\(a\) kwa\(n\) mara yenyewe.

    \[a^n= a\cdot a \cdot a\cdot a\cdot a\cdot a\cdot a\cdot a \ldots \ldots \cdot a \nonumber \]

    Nukuu:

    \(a\)ni msingi,\(n\) ni exponent chanya.

    \(a^n\)ni kusoma kama”\(a\) alimfufua kwa nguvu ya\(n\).”

    Kutambua msingi na exponent katika maneno.

    \(2^4\)\(x^5\),\(\left(\dfrac{3}{7}\right)^7\),\((-3)^3\)

    Suluhisho
    Ufafanuzi Msingi Mtetezi
    \(2^4\) 2 4
    \(x^5\) \(x\) 5
    \(\left(\dfrac{3}{7}\right)^7\) \(\dfrac{3}{7}\) 7
    \((-3)^3\) -3 3
    Tatizo la mazoezi

    Kutambua msingi na exponent ya yafuatayo.

    Ufafanuzi Msingi Mtetezi
    \(7^9\)
    \((-11)^6\)
    \(a^b\)
    \(\left(\dfrac{11}{12}\right)^5\)
    \(12^3\)
    \(\left(-\dfrac{7}{3}\right)^2\)
    \(x^7\)
    \((2.56)^4\)

    Kutathmini maneno ya fomu\(a^n\)

    Wakati msingi na exponent ni thamani namba inawezekana kutathmini usemi imeandikwa katika na exponent. Ili kupata thamani, tumia ufafanuzi na kupanua maneno. Mara baada ya kupanua, kuzidisha na matokeo ni thamani ya namba ya kujieleza.

    Panua maneno yafuatayo na tathmini ikiwa inawezekana.

    \(3^4\),\(\left(\dfrac{3}{5}\right)^3\)\(x^7\),\((3.12)^2\),\((-5)^3\),\((-y)^6\)

    Suluhisho
    \(3^4\) \(= 3\cdot 3\cdot 3\cdot 3 = 81\)
    \(\left(\dfrac{3}{5}\right)^3\) \(\dfrac{3 }{5} \cdot \dfrac{3}{ 5 }\cdot \dfrac{3 }{5} = \dfrac{27 }{125}\)
    \(x^7\)

    \(x\cdot x\cdot x\cdot x\cdot x\cdot x\cdot x\)

    Kumbuka: Haiwezi kutathmini tangu x haijulikani

    \((3.12)^2\) \((3.12)\cdot (3.12) = 9.734\)
    \((-5)^3\) \(−5 \cdot −5 \cdot −5 = −12\)
    \((-y)^6\)

    \(−y \cdot −y \cdot −y \cdot −y \cdot −y \cdot −y = y^6\)

    Kumbuka: y haijulikani

    Panua maneno yafuatayo na tathmini ikiwa inawezekana.

    1. \(7^3\)
    2. \(\left(−\dfrac{ 2 }{3}\right)^4\)
    3. \((−x)^7\)
    4. \((7.14)^2\)
    5. \((−3)^9\)
    6. \((z)^5\)
    7. \(\left(− \dfrac{11 }{33 }\right)^2\)
    8. \(6^5\)
    9. \(\left(\dfrac{x}{ y}\right)^4\)
    10. \(a^{10}\)
    11. \(\left(\dfrac{2}{x}\right)^3\)