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 $u^4\cdot \sqrt{u}$.