Question

simplify.
\sqrt{w^{15}}
assume that the variable represents a positive real number.

Explanation:

Step1: Rewrite the exponent

We can rewrite \( w^{15} \) as \( w^{14} \times w \), since \( 14 + 1 = 15 \). So, \( \sqrt{w^{15}}=\sqrt{w^{14}\times w} \).

Step2: Use the property of square roots

Recall that \( \sqrt{ab}=\sqrt{a}\times\sqrt{b} \) (for \( a\geq0,b\geq0 \)). So, \( \sqrt{w^{14}\times w}=\sqrt{w^{14}}\times\sqrt{w} \).

Step3: Simplify \( \sqrt{w^{14}} \)

Using the property \( \sqrt{x^{2n}} = x^{n} \) (for \( x\geq0 \)), here \( x = w \) and \( 2n = 14 \), so \( n = 7 \). Thus, \( \sqrt{w^{14}}=w^{7} \).

Step4: Combine the terms

Putting it all together, \( \sqrt{w^{14}}\times\sqrt{w}=w^{7}\sqrt{w} \).

Answer:

\( w^{7}\sqrt{w} \)