Question

simplify.
\sqrt{u^{20}}
assume that the variable represents a positive real number.

Explanation:

Step1: Recall the property of square roots for exponents

For a positive real number \( a \) and integer \( n \), \( \sqrt{a^{m}}=a^{\frac{m}{2}} \) when \( m \) is even (since \( a \) is positive, we don't have to worry about principal roots issues here). So for \( \sqrt{u^{20}} \), we can apply the formula \( \sqrt{x^{n}} = x^{\frac{n}{2}} \) where \( x = u \) and \( n = 20 \).

Step2: Calculate the exponent

Substitute \( n = 20 \) into \( \frac{n}{2} \), we get \( \frac{20}{2}=10 \). So \( \sqrt{u^{20}}=u^{10} \) because \( u \) is a positive real number, and raising it to the 10th power is valid.

Answer:

\( u^{10} \)