On simplification, \(\frac{(625)^{6.25}\times (\sqrt5)^{10.4}}{(\sqrt5)^{54}\times (5)^{1.2}\times (25)^{0.5}}\) reduces to:

Given:

Expression: \( \frac{(625)^{6.25} × (\sqrt5)^{10.4}}{(\sqrt5)^{54} × 5^{1.2} × (25)^{0.5}} \)

Formula used:

Power of a power: \((a^m)^n = a^{m×n}\)

Roots as fractional exponents: \(\sqrt5=5^{1/2},\;25=5^2\)

Calculations:

⇒ \(625=5^4\)

⇒ \((5^4)^{6.25}=5^{4×6.25}=5^{25}\)

⇒ \((\sqrt5)^{10.4}=(5^{1/2})^{10.4}=5^{10.4/2}=5^{5.2}\)

⇒ Numerator = \(5^{25}×5^{5.2}=5^{30.2}\)

⇒ \((\sqrt5)^{54}=5^{54/2}=5^{27}\)

⇒ \((25)^{0.5}=(5^2)^{0.5}=5^{2×0.5}=5^1\)

⇒ Denominator exponent = 27 + 1.2 + 1 = 29.2 ⇒ Denominator = \(5^{29.2}\)

⇒ Fraction = \(5^{30.2−29.2}=5^1\) = 5

∴ The expression simplifies to 5.