Simplify: \(\dfrac{5^{3x} \times (5^{2})^{x-1}}{5^{4x-1}}\)
\(5^{x-1}\)
Rewrite \((5^{2})^{x-1} = 5^{2(x-1)} = 5^{2x-2}\).
Multiply the numerator terms by adding exponents: \(5^{3x} \times 5^{2x-2} = 5^{3x + 2x - 2} = 5^{5x-2}\).
Divide by \(5^{4x-1}\) by subtracting exponents: \(5^{(5x-2) - (4x-1)}\).
Simplify the exponent: \((5x - 2) - (4x - 1) = 5x - 2 - 4x + 1 = x - 1\).
So the expression equals \(5^{x-1}\).
Hence, the simplified value is \(5^{x-1}\).
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