To rewrite the expression \( 125^{3/2} \) as a product of identical factors, we can break it down as follows:
First, we recognize that \( 125 \) can be expressed as \( 5^3 \). Therefore, we have:
\[ 125^{3/2} = (5^3)^{3/2} \]
Now, using the property of exponents \( (a^m)^n = a^{m \cdot n} \):
\[ (5^3)^{3/2} = 5^{3 \cdot \frac{3}{2}} = 5^{\frac{9}{2}} \]
Next, we can rewrite \( 5^{\frac{9}{2}} \) in terms of product of identical factors. Notably, \( 5^{\frac{9}{2}} = 5^{4.5} = 5^4 \cdot 5^{1/2} \):
\[ 5^{\frac{9}{2}} = 5^4 \cdot 5^{1/2} \]
Calculating \( 5^4 \):
\[ 5^4 = 625 \]
And \( 5^{1/2} = \sqrt{5} \). Therefore, we have:
\[ 125^{3/2} = 625 \cdot \sqrt{5} \]
Thus, the final result in simplified form is:
\[ 125^{3/2} = 625\sqrt{5} \]
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