Which of the following expressions is equivalent to the radical expression ^3 square root m^2n^5 explain answer
m^3/2n^3/5
m^2/3n^5/3
m^5n^8
mn^2
The expression that is equivalent to the given radical expression is mn^2.
To understand why, let's first simplify the given radical expression. The cube root (∛) of (m^2n^5) can be rewritten as (m^(2/3))(n^(5/3)). This is because when you take the cube root of a number raised to a power, you can distribute the root to each of the exponents.
Now, let's compare the simplified expression (m^(2/3))(n^(5/3)) to the answer choices:
- m^(3/2)n^(3/5): This expression does not match the simplified expression. The exponents are different, so it is not equivalent.
- m^(2/3)n^(5/3): This expression matches the simplified expression. The exponents are the same, so it is equivalent.
- m^5n^8: This expression does not match the simplified expression. The exponents are different, so it is not equivalent.
- mn^2: This expression does not match the simplified expression. The exponents are different, so it is not equivalent.
Therefore, the only expression that is equivalent to the given radical expression is m^(2/3)n^(5/3), or option (B).