To express [tex]\( 1250^{\frac{3}{4}} \)[/tex] in its simplest radical form, let's break the problem into smaller steps.
1. Prime Factorization:
Start with the prime factorization of 1250.
[tex]\[
1250 = 125 \times 10 = 5^3 \times 2 \times 5 = 5^4 \times 2
\][/tex]
2. Exponentiation:
We have the expression [tex]\( 1250^{\frac{3}{4}} \)[/tex]. Using our factorization, we can write:
[tex]\[
(5^4 \times 2)^{\frac{3}{4}}
\][/tex]
3. Distribute the exponent:
Apply the exponent to each part separately:
[tex]\[
(5^4)^{\frac{3}{4}} \times 2^{\frac{3}{4}}
\][/tex]
4. Simplify each term:
Simplify each part of the expression:
[tex]\[
(5^4)^{\frac{3}{4}} = 5^{4 \times \frac{3}{4}} = 5^3
\][/tex]
And,
[tex]\[
2^{\frac{3}{4}}
\][/tex]
5. Combine the results:
Combine the simplified terms back together:
[tex]\[
5^3 \times 2^{\frac{3}{4}}
\][/tex]
So, the expression [tex]\( 1250^{\frac{3}{4}} \)[/tex] in its simplest radical form can be written as:
[tex]\[
5^3 \sqrt[4]{2^3}
\][/tex]
In this form, the value that remains under the radical is associated with the term [tex]\( 2^{3/4} \)[/tex], and it involves the number 2.
Thus, the value that remains under the radical is:
[tex]\[
\boxed{2}
\][/tex]
