Certainly! Let's break down the given expression step by step.
We start with the given mathematical expression:
[tex]\[ 3d^2n + f \cdot 4 \times \sqrt{27d^6n} + 64 \cdot f^{12y} \][/tex]
### Step 1: Simplify Each Term
#### Term 1: [tex]\( 3d^2n \)[/tex]
This term is already in its simplest form:
[tex]\[ 3d^2n \][/tex]
#### Term 2: [tex]\( f \cdot 4 \times \sqrt{27d^6n} \)[/tex]
Firstly, simplify inside the square root:
[tex]\[ \sqrt{27d^6n} \][/tex]
[tex]\[ = \sqrt{27d^6 \cdot n} \][/tex]
Notice that [tex]\( 27d^6 \)[/tex] can be further simplified because [tex]\( 27 = 3^3 \)[/tex]:
[tex]\[ = \sqrt{(3^3)d^6 \cdot n} \][/tex]
[tex]\[ = \sqrt{3^3 \cdot d^6 \cdot n} \][/tex]
Since [tex]\( 3^3 = 27 \)[/tex] and the square root of [tex]\( d^6 \)[/tex] is [tex]\( d^3 \)[/tex], we can rewrite:
[tex]\[ = \sqrt{27} \cdot \sqrt{d^6} \cdot \sqrt{n} \][/tex]
[tex]\[ = 3\sqrt{3} \cdot d^3 \cdot \sqrt{n} \][/tex]
We plug this back into the term:
[tex]\[ f \cdot 4 \times 3\sqrt{3} \cdot d^3 \cdot \sqrt{n} \][/tex]
[tex]\[ = f \cdot 12\sqrt{3} \cdot d^3 \cdot \sqrt{n} \][/tex]
[tex]\[ = 12\sqrt{3} \cdot f \cdot d^3 \cdot \sqrt{n} \][/tex]
#### Term 3: [tex]\( 64 \cdot f^{12y} \)[/tex]
This term is already in its simplest form:
[tex]\[ 64f^{12y} \][/tex]
### Step 2: Combine the Simplified Terms
Combining the three simplified terms, we obtain:
[tex]\[ 3d^2n + 12\sqrt{3}f\sqrt{d^6n} + 64f^{12y} \][/tex]
### Final Answer
Therefore, the detailed expression can be written as:
[tex]\[ 3d^2n + 12\sqrt{3}f\sqrt{d^6n} + 64f^{12y} \][/tex]
This is the simplified form of the given mathematical expression.
