Let's simplify the expression [tex]\( 10 \sqrt{147} + \sqrt{27} \)[/tex] step-by-step and express it in the simplest radical form.
### Step 1: Simplify [tex]\( 10 \sqrt{147} \)[/tex]
First, we need to simplify [tex]\( \sqrt{147} \)[/tex].
Notice that:
[tex]\[ 147 = 3 \cdot 49 \][/tex]
[tex]\[ \sqrt{147} = \sqrt{3 \cdot 49} = \sqrt{3 \cdot 7^2} \][/tex]
Since [tex]\( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \)[/tex], we get:
[tex]\[ \sqrt{147} = \sqrt{3} \cdot \sqrt{49} = \sqrt{3} \cdot 7 = 7 \sqrt{3} \][/tex]
So:
[tex]\[ 10 \sqrt{147} = 10 \cdot 7 \sqrt{3} = 70 \sqrt{3} \][/tex]
### Step 2: Simplify [tex]\( \sqrt{27} \)[/tex]
Now, let's simplify [tex]\( \sqrt{27} \)[/tex].
Notice that:
[tex]\[ 27 = 3^3 \][/tex]
[tex]\[ \sqrt{27} = \sqrt{3^3} = \sqrt{3 \cdot 9} = \sqrt{3} \cdot \sqrt{9} = \sqrt{3} \cdot 3 = 3 \sqrt{3} \][/tex]
### Step 3: Combine the Simplified Terms
We now combine the simplified terms:
[tex]\[ 10 \sqrt{147} + \sqrt{27} = 70 \sqrt{3} + 3 \sqrt{3} \][/tex]
Since both terms have the common factor [tex]\( \sqrt{3} \)[/tex], we can combine them:
[tex]\[ 70 \sqrt{3} + 3 \sqrt{3} = (70 + 3) \sqrt{3} = 73 \sqrt{3} \][/tex]
### Final Answer
Therefore, the expression [tex]\( 10 \sqrt{147} + \sqrt{27} \)[/tex] in its simplest radical form is:
[tex]\[
\boxed{73 \sqrt{3}}
\][/tex]
