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Find the value of m, satisfying the equation: (232)3×(223)14×(232)5=(223)6m+9\left(\frac{23}{2}\right)^3 \times \left(\frac{2}{23}\right)^{14} \time
Question

Find the value of m, satisfying the equation:
(232)3×(223)14×(232)5=(223)6m+9\left(\frac{23}{2}\right)^3 \times \left(\frac{2}{23}\right)^{14} \times \left(\frac{23}{2}\right)^5=\left(\frac{2}{23}\right)^{6m+9}

A.

-3/6

B.

3/6

C.

-10/6

D.

-12/6

Correct option is A

Given
Equation: (232)3×(223)14×(232)5=(223)6m+9\left(\frac{23}{2}\right)^3 \times \left(\frac{2}{23}\right)^{14} \times \left(\frac{23}{2}\right)^5=\left(\frac{2}{23}\right)^{6m+9}​​

Formula Used
am×an=am+n am=1ama^m \times a^n = a^{m+n} \\\ \\a^{-m} = \frac{1}{a^m}​​

Solution
(232)3×(223)14×(232)5=(223)6m+9=(223)3×(223)14×(223)5=(223)6m+9=(223)11×(223)5=(223)6m+9=(223)6=(223)6m+9=6=6m+9=3=6m=m=36=12\left(\frac{23}{2}\right)^3 \times \left(\frac{2}{23}\right)^{14} \times \left(\frac{23}{2}\right)^5=\left(\frac{2}{23}\right)^{6m+9}\\=\left(\frac{2}{23}\right)^{-3}\times\left(\frac{2}{23}\right)^{14}\times\left(\frac{2}{23}\right)^{-5}=\left(\frac{2}{23}\right)^{6m+9}\\=\left(\frac{2}{23}\right)^{11}\times\left(\frac{2}{23}\right)^{-5}=\left(\frac{2}{23}\right)^{6m+9}\\=\left(\frac{2}{23}\right)^6=\left(\frac{2}{23}\right)^{6m+9}\\=6=6m+9\\=-3=6m\\=m= -\frac36= -\frac12​​

Final Answer
So the correct answer is (a)

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