Solving for m in Exponential Equation
The problem requires finding the value of 'm' in the given exponential equation:
$ \frac{9^m \times 3^5 \times 27^3}{3 \times 81^4} = 3^9 $To solve this, we express all the numbers as powers of the base 3.
- $9 = 3^2$
- $27 = 3^3$
- $81 = 3^4$
Simplifying the Equation
Substitute these powers back into the original equation:
$ \frac{(3^2)^m \times 3^5 \times (3^3)^3}{3^1 \times (3^4)^4} = 3^9 $Apply the power of a power rule, $(a^b)^c = a^{bc}$:
$ \frac{3^{2m} \times 3^5 \times 3^{9}}{3^1 \times 3^{16}} = 3^9 $Use the product of powers rule, $a^b \times a^c = a^{b+c}$, for the numerator:
$ \frac{3^{2m + 5 + 9}}{3^{1 + 16}} = 3^9 $ $ \frac{3^{2m + 14}}{3^{17}} = 3^9 $Apply the division of powers rule, $\frac{a^b}{a^c} = a^{b-c}$:
$ 3^{(2m + 14) - 17} = 3^9 $ $ 3^{2m - 3} = 3^9 $Finding the Value of m
Now, equate the exponents since the bases are the same:
$ 2m - 3 = 9 $Solve the linear equation for 'm':
- Add 3 to both sides: $2m = 9 + 3$
- $2m = 12$
- Divide by 2: $m = \frac{12}{2}$
- $m = 6$
Thus, the value of m is 6.
Similar Questions
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Important Questions from Surds and Indices
The value of (0.3) [{(200 - 146)/(3 × 3 × 3)} - 3] is:
The expression \(\frac{{15\left( {\sqrt {10} + \sqrt 5 } \right)}}{{\sqrt {10\;} + \sqrt {20} + \sqrt {40} - \sqrt 5 - \sqrt {80} }}\) is equal to:
Let \(x = \left( {\frac{{√ {1875} }}{{√ {3888} }} \div \frac{{√ {1200} }}{{\sqrt 768}}} \right) \times \frac{{√ {175} }}{{√ {1792} }}\) . Then √x is equal to:
If \(x = \sqrt {-\sqrt 3 + \sqrt {3 + 8\sqrt {7 + 4\sqrt 3}}}\) where x > 0, then the value of x is equal to:
What is the value of \(\frac{\sqrt{7}+\sqrt{5}}{\sqrt{7}−\sqrt{5}} \div \frac{\sqrt{14}+\sqrt{10}}{\sqrt{14}−\sqrt{10}}+\frac{\sqrt{10}}{\sqrt{5}}\) ?
