Simplify square root of u^9

Solution:

Step 1:

Express $u^{9}$ as the product of $u^{8}$ and $u$: $u^{9} = u^{8} \cdot u$.

Step 1.1:

Isolate $u^{8}$ from the expression: $\sqrt{u^{8} \cdot u}$.

Step 1.2:

Represent $u^{8}$ as $(u^{4})^{2}$: $\sqrt{(u^{4})^{2} \cdot u}$.

Step 2:

Extract $u^{4}$ from the square root, as it is a perfect square: $u^{4} \cdot \sqrt{u}$.

Knowledge Notes:

To simplify the square root of a variable raised to a power, such as $\sqrt{u^{9}}$, we can use the properties of exponents and radicals. Here are the relevant knowledge points:

1. Exponent Rules: For any non-negative real number $a$ and integers $m$ and $n$:

   • $a^{m} \cdot a^{n} = a^{m+n}$
   • $(a^{m})^{n} = a^{m \cdot n}$
   • $a^{m/n} = \sqrt[n]{a^{m}}$ (where $n$ is the root)

2. Square Roots and Perfect Squares: The square root of a perfect square, such as $a^{2}$, is simply $a$. This is because $\sqrt{a^{2}} = a$.

3. Simplifying Radicals: When simplifying the square root of a variable raised to an even power, we can rewrite the variable as a perfect square raised to a power. For example, $u^{8}$ can be written as $(u^{4})^{2}$, which simplifies to $u^{4}$ when taken out of the square root.

4. Odd Exponents: When dealing with odd exponents, we can factor out the largest even power and simplify the remaining factor under the radical. For example, $u^{9}$ can be factored into $u^{8} \cdot u$, where $u^{8}$ is the largest even power that can be simplified.

By applying these rules, we can simplify the expression $\sqrt{u^{9}}$ step by step, ultimately arriving at $u^{4} \cdot \sqrt{u}$.