Answer :
Let's start by simplifying the expression [tex]\( 1250^{\frac{3}{4}} \)[/tex] step-by-step and finding out which value remains under the radical in its simplest form.
1. Interpret the Exponent [tex]\( \frac{3}{4} \)[/tex]:
- The exponent [tex]\( \frac{3}{4} \)[/tex] can be broken into two parts:
[tex]\[ 1250^{\frac{3}{4}} = (1250^{\frac{1}{4}})^3 \][/tex]
- This means we first need to find the fourth root of 1250 and then raise the result to the power of 3.
2. Calculate the Fourth Root:
- Compute the fourth root of 1250 (i.e., [tex]\( 1250^{\frac{1}{4}} \)[/tex]). This value is approximately:
[tex]\[ 1250^{\frac{1}{4}} \approx 5.946 \][/tex]
3. Raise the Fourth Root to the Power of 3:
- Next, we need to raise the fourth root result to the power of 3:
[tex]\[ (5.946)^3 \approx 210.224 \][/tex]
4. Simplify the Expression Further:
- To further simplify, we recognize that the expression under the radical can sometimes be simplified to an integer under a different radical form. Here, we want to find the simplest radical form to understand what remains under the radical.
5. Identify Remaining Value Under Radical:
- The cube root (since the exponent had 3 in the numerator) of [tex]\( 210.224 \)[/tex] simplifies to a number that retains a radical. In this case, the remaining value to consider is:
[tex]\[ \sqrt[3]{210.224} \approx 6 \][/tex]
Therefore, when [tex]\( 1250^{\frac{3}{4}} \)[/tex] is expressed in its simplest radical form, the value that remains under the radical is 6.
Thus, the correct answer is:
[tex]\[ \boxed{6} \][/tex]
1. Interpret the Exponent [tex]\( \frac{3}{4} \)[/tex]:
- The exponent [tex]\( \frac{3}{4} \)[/tex] can be broken into two parts:
[tex]\[ 1250^{\frac{3}{4}} = (1250^{\frac{1}{4}})^3 \][/tex]
- This means we first need to find the fourth root of 1250 and then raise the result to the power of 3.
2. Calculate the Fourth Root:
- Compute the fourth root of 1250 (i.e., [tex]\( 1250^{\frac{1}{4}} \)[/tex]). This value is approximately:
[tex]\[ 1250^{\frac{1}{4}} \approx 5.946 \][/tex]
3. Raise the Fourth Root to the Power of 3:
- Next, we need to raise the fourth root result to the power of 3:
[tex]\[ (5.946)^3 \approx 210.224 \][/tex]
4. Simplify the Expression Further:
- To further simplify, we recognize that the expression under the radical can sometimes be simplified to an integer under a different radical form. Here, we want to find the simplest radical form to understand what remains under the radical.
5. Identify Remaining Value Under Radical:
- The cube root (since the exponent had 3 in the numerator) of [tex]\( 210.224 \)[/tex] simplifies to a number that retains a radical. In this case, the remaining value to consider is:
[tex]\[ \sqrt[3]{210.224} \approx 6 \][/tex]
Therefore, when [tex]\( 1250^{\frac{3}{4}} \)[/tex] is expressed in its simplest radical form, the value that remains under the radical is 6.
Thus, the correct answer is:
[tex]\[ \boxed{6} \][/tex]
