To simplify [tex]\(\sqrt{50 u^{12} v^8}\)[/tex], let's break down the expression step by step.
### Step 1: Factor the expression inside the square root
The expression inside the square root is [tex]\(50 u^{12} v^8\)[/tex]. We can factor this as the product of simpler expressions:
[tex]\[
50 u^{12} v^8 = 50 \times u^{12} \times v^8
\][/tex]
### Step 2: Simplify the square root of a constant
The square root of 50 can be simplified by recognizing that 50 can be written as [tex]\(25 \times 2\)[/tex]:
[tex]\[
\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5 \sqrt{2}
\][/tex]
### Step 3: Simplify the square root of the variable parts
Next, we handle the variable parts [tex]\(u^{12}\)[/tex] and [tex]\(v^8\)[/tex].
- For [tex]\(u^{12}\)[/tex], the square root can be simplified because [tex]\(\sqrt{u^{12}} = u^{12/2} = u^6\)[/tex].
- For [tex]\(v^8\)[/tex], the square root can be simplified because [tex]\(\sqrt{v^8} = v^{8/2} = v^4\)[/tex].
### Step 4: Combine the simplified parts
Now, bringing all simplified parts together, we have:
[tex]\[
\sqrt{50 u^{12} v^8} = 5 \sqrt{2} \times u^6 \times v^4
\][/tex]
So, the simplified form of [tex]\(\sqrt{50 u^{12} v^8}\)[/tex] is:
[tex]\[
5 \sqrt{2} u^6 v^4
\][/tex]
