Solving Exponential Equation for x
We are given the equation: $5^{3x+4} = (5^4)^{105}$
Simplifying the Equation
Using the exponent rule $ (a^m)^n = a^{m \times n} $, we simplify the right side:
$ (5^4)^{105} = 5^{4 \times 105} = 5^{420} $The equation now becomes:
$ 5^{3x+4} = 5^{420} $Equating Exponents
Since the bases are the same (base 5), the exponents must be equal:
$ 3x + 4 = 420 $Solving for x
To find the value of $x$, we first isolate the term with $x$ by subtracting 4 from both sides:
$ 3x = 420 - 4 $ $ 3x = 416 $Now, we divide by 3 to solve for $x$:
$ x = \frac{416}{3} $ $ x \approx 138.666... $Finding the Integral Part
The integral part of a number is the whole number part before the decimal point. For $x \approx 138.666...$, the integral part is 138.
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 \(\frac{\sqrt{7}+\sqrt{5}}{\sqrt{7}−\sqrt{5}} \div \frac{\sqrt{14}+\sqrt{10}}{\sqrt{14}−\sqrt{10}}+\frac{\sqrt{10}}{\sqrt{5}}\) ?
