Question

simplify.
$5\sqrt{w^{17}} + 17w^{8}\sqrt{w}$
assume that the variable represents a positive real number.

Explanation:

Step1: Simplify \( \sqrt{w^{17}} \)

We can rewrite \( w^{17} \) as \( w^{16} \times w \) (since \( 16 + 1 = 17 \)). Then, using the property of square roots \( \sqrt{ab}=\sqrt{a}\times\sqrt{b} \) (where \( a = w^{16} \) and \( b = w \)) and \( \sqrt{w^{16}}=w^{8} \) (because \( (w^{8})^2 = w^{16} \) and \( w \) is positive), we get:
\( \sqrt{w^{17}}=\sqrt{w^{16}\times w}=\sqrt{w^{16}}\times\sqrt{w}=w^{8}\sqrt{w} \)
So, \( 5\sqrt{w^{17}} = 5w^{8}\sqrt{w} \)

Step2: Combine like terms

Now we have the expression \( 5w^{8}\sqrt{w}+17w^{8}\sqrt{w} \). Since both terms have \( w^{8}\sqrt{w} \), we can factor that out:
\( (5 + 17)w^{8}\sqrt{w} \)
Calculating \( 5+17 = 22 \), we get:
\( 22w^{8}\sqrt{w} \)

Answer:

\( 22w^{8}\sqrt{w} \)