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How to solve the equation m^m5 = 100?

Mathematics

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Answer

m=105m = \sqrt[5]{10}

Step-by-step solution

6 steps

برای حل معادله mm5=100m^{m^5} = 100، مراحل زیر را دنبال می‌کنیم:

  1. معادله اصلی را بنویسید.

    mm5=100m^{m^5} = 100

  2. هر دو طرف معادله را به توان 5 برسانید.

    (mm5)5=1005(m^{m^5})^5 = 100^5

  3. از خاصیت (ab)c=abc(a^b)^c = a^{bc} استفاده کنید.

    (m5)m5=1005(m^5)^{m^5} = 100^5

  4. عدد 100 را به صورت 10210^2 بنویسید و سپس توان را اعمال کنید.

    (m5)m5=(102)5(m^5)^{m^5} = (10^2)^5 (m5)m5=102×5(m^5)^{m^5} = 10^{2 \times 5} (m5)m5=1010(m^5)^{m^5} = 10^{10}

  5. با مقایسه دو طرف معادله، مقدار m5m^5 را پیدا کنید.

    اگر XX=YYX^X = Y^Y و X,Y>0X, Y > 0 باشد، آنگاه X=YX=Y.
    در اینجا، X=m5X = m^5 و Y=10Y = 10.
    پس:

    m5=10m^5 = 10

  6. برای یافتن mm، از هر دو طرف ریشه پنجم بگیرید.

    m=105m = \sqrt[5]{10}
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