When 1,250^3/4 is written in simplest radical form, which value remains under the radical?
2
5
6
8
The Correct Answer and Explanation is :
To simplify the expression ( 1250^{3/4} ) into its simplest radical form, we need to follow a series of steps involving exponent rules and radical simplification.
Step 1: Rewrite the expression with a fractional exponent
The expression is given as ( 1250^{3/4} ). According to the rules of exponents, ( a^{m/n} ) is the same as the ( n )-th root of ( a^m ), which can also be written as ( \sqrt[n]{a^m} ).
Thus, we rewrite ( 1250^{3/4} ) as:
[
1250^{3/4} = \left(1250^{1/4}\right)^3
]
This expression tells us that we first take the fourth root of 1250 and then cube the result.
Step 2: Simplify the fourth root of 1250
Now we need to simplify ( 1250^{1/4} ). To do this, we break 1250 into its prime factors:
[
1250 = 2 \times 5^4
]
This factorization is helpful because we can now take the fourth root of each factor separately. Specifically, the fourth root of ( 5^4 ) is ( 5 ), and the fourth root of ( 2 ) remains as ( 2^{1/4} ).
Thus, the fourth root of 1250 is:
[
1250^{1/4} = 5 \times 2^{1/4}
]
Step 3: Cube the result
Next, we cube the result of ( 1250^{1/4} ):
[
\left(5 \times 2^{1/4}\right)^3 = 5^3 \times \left(2^{1/4}\right)^3
]
Simplifying further:
[
5^3 = 125 \quad \text{and} \quad \left(2^{1/4}\right)^3 = 2^{3/4}
]
So, the expression becomes:
[
1250^{3/4} = 125 \times 2^{3/4}
]
Step 4: Final simplified form
The simplified radical form of ( 1250^{3/4} ) is:
[
1250^{3/4} = 125 \times 2^{3/4}
]
In this form, ( 2^{3/4} ) is the part of the expression that remains under the radical.
Final Answer:
The correct value that remains under the radical is 2.