Question

fill in the blank.
the expressions (sqrt4{10mn}) and ((10mn)^{\frac{1}{4}}) are ______.
the expressions (sqrt4{10mn}) and ((10mn)^{\frac{1}{4}}) are (\boxed{\text{dropdown}}).
dropdown options: not equivalent, equivalent

Explanation:

Step1: Recall the radical exponent rule

The rule for converting a radical to an exponent is that the \(n\)-th root of a number \(a\), \(\sqrt[n]{a}\), can be written as \(a^{\frac{1}{n}}\).

Step2: Apply the rule to the given radical

For the expression \(\sqrt[4]{10mn}\), using the rule \(\sqrt[n]{a}=a^{\frac{1}{n}}\) where \(n = 4\) and \(a=10mn\), we get \(\sqrt[4]{10mn}=(10mn)^{\frac{1}{4}}\).

Step3: Compare the two expressions

We have the first expression \(\sqrt[4]{10mn}\) which we just showed is equal to \((10mn)^{\frac{1}{4}}\) and the second expression is \((10mn)^{\frac{1}{4}}\). So they are equivalent.

Answer:

equivalent