Question

simplify.
\sqrt{v^{18}}
assume that the variable represents a positive real number.

Explanation:

Step1: Recall the property of square roots and exponents

For a positive real number \( a \) and integer \( n \), \( \sqrt{a^m}=a^{\frac{m}{2}} \) (when \( m \) is even, and \( a\geq0 \)). Here we have \( \sqrt{v^{18}} \), so we can apply the rule \( \sqrt{x}=x^{\frac{1}{2}} \) and then use the power - of - a - power rule \( (x^m)^n=x^{m\times n} \).
First, rewrite \( \sqrt{v^{18}} \) as \( (v^{18})^{\frac{1}{2}} \).

Step2: Apply the power - of - a - power rule

Using the formula \( (x^m)^n = x^{m\times n} \), where \( x = v \), \( m = 18 \), and \( n=\frac{1}{2} \). Then \( (v^{18})^{\frac{1}{2}}=v^{18\times\frac{1}{2}} \).
Calculate \( 18\times\frac{1}{2}=9 \). So \( v^{18\times\frac{1}{2}} = v^{9} \).

Answer:

\( v^{9} \)