Math Snap

STEP 1

1. The expression to simplify is $\sqrt{16 u^{9}}$.
2. The variable $u$ represents a positive real number.
3. Simplifying involves expressing the radical in terms of integer exponents.

STEP 2

1. Simplify the numerical part of the radical.
2. Simplify the variable part of the radical.

STEP 3

Simplify the numerical part of the radical $\sqrt{16}$.
$$\sqrt{16} = 4$$

STEP 4

Simplify the variable part of the radical $\sqrt{u^9}$.
First, express $u^9$ as a power suitable for simplification:
$$u^9 = (u^4)^2 \times u$$ This allows us to use the property of radicals $\sqrt{a^2} = a$.

STEP 5

Apply the property of radicals to simplify $\sqrt{(u^4)^2 \times u}$:
$$\sqrt{(u^4)^2 \times u} = u^4 \times \sqrt{u}$$

Combine the simplified numerical and variable parts:
$$\sqrt{16 u^9} = 4 \times u^4 \times \sqrt{u}$$ The simplified expression is:
$$\boxed{4u^4\sqrt{u}}$$