To solve the expression [tex]\( \sqrt[2]{25^3} \)[/tex], we first need to understand how to handle fractional exponents. The expression [tex]\( \sqrt[2]{25^3} \)[/tex] can be rewritten using exponent notation as [tex]\( 25^{\frac{3}{2}} \)[/tex].
Now let's understand why this equivalence holds:
- [tex]\( 25^3 \)[/tex] means we raise 25 to the power of 3: [tex]\( 25 \times 25 \times 25 \)[/tex].
- Taking the square root of a number is the same as raising it to the power of [tex]\(\frac{1}{2}\)[/tex]: [tex]\( \sqrt[2]{a} = a^{\frac{1}{2}} \)[/tex].
So combining these operations, the expression becomes:
[tex]\[ \sqrt[2]{25^3} = (25^3)^{\frac{1}{2}} \][/tex]
Using the property of exponents [tex]\((a^m)^n = a^{mn}\)[/tex], this can be rewritten as:
[tex]\[ (25^3)^{\frac{1}{2}} = 25^{3 \times \frac{1}{2}} = 25^{\frac{3}{2}} \][/tex]
Therefore, the expression [tex]\( \sqrt[2]{25^3} \)[/tex] simplifies to [tex]\( 25^{\frac{3}{2}} \)[/tex].
The correct answer is:
[tex]\[ 25^{\frac{3}{2}} \][/tex]
