Certainly! Let's simplify the given expression [tex]\(\sqrt{27 u^{10}}\)[/tex] step-by-step:
### Step 1: Simplify the expression under the square root
The given expression is:
[tex]\[ \sqrt{27 u^{10}} \][/tex]
Notice that we can break down the expression inside the square root into a product of simpler terms:
[tex]\[ 27 u^{10} = 27 \cdot u^{10} \][/tex]
### Step 2: Factorize 27
The number 27 can be factorized as:
[tex]\[ 27 = 3^3 \][/tex]
So the expression inside the square root becomes:
[tex]\[ 27 u^{10} = 3^3 \cdot u^{10} \][/tex]
### Step 3: Apply the square root to the product
We can use the property of square roots that [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex] to separate the terms:
[tex]\[ \sqrt{27 u^{10}} = \sqrt{3^3 \cdot u^{10}} = \sqrt{3^3} \cdot \sqrt{u^{10}} \][/tex]
### Step 4: Simplify each square root
For [tex]\(\sqrt{3^3}\)[/tex], we can simplify as follows:
[tex]\[ \sqrt{3^3} = \sqrt{3 \cdot 3^2} = \sqrt{3} \cdot \sqrt{3^2} = \sqrt{3} \cdot 3 = 3 \cdot \sqrt{3} \][/tex]
For [tex]\(\sqrt{u^{10}}\)[/tex], we use the property [tex]\(\sqrt{u^{2k}} = u^k\)[/tex] where [tex]\(k\)[/tex] is an integer:
[tex]\[ \sqrt{u^{10}} = \sqrt{(u^5)^2} = u^5 \][/tex]
### Step 5: Combine the results
Putting these simplified parts together, we get:
[tex]\[ \sqrt{27 u^{10}} = 3 \cdot \sqrt{3} \cdot u^5 \][/tex]
### Final Answer
So the simplified form of the given expression is:
[tex]\[ \sqrt{27 u^{10}} = 3 \sqrt{3} \cdot u^5 \][/tex]
In conclusion, the expression [tex]\(\sqrt{27 u^{10}}\)[/tex] simplifies to:
[tex]\[ 3 \sqrt{3} u^5 \][/tex]
