To simplify the expressions provided, we will work through each expression step by step.
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Expression: \( f^2 g^3 h^4 \sqrt{f^3 g} \)
First, simplify the square root: \[ \sqrt{f^3 g} = \sqrt{f^3} \cdot \sqrt{g} = f^{3/2} g^{1/2} \] Now substituting back into the expression: \[ f^2 g^3 h^4 \cdot (f^{3/2} g^{1/2}) = f^{2 + 3/2} g^{3 + 1/2} h^4 = f^{4.5} g^{3.5} h^4 \] Therefore, this expression simplifies to: \[ f^{9/2} g^{7/2} h^4 \]
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Expression: \( f^3 g^5 h^8 \sqrt{f} \)
Again, simplify the square root: \[ \sqrt{f} = f^{1/2} \] Now substituting back: \[ f^3 g^5 h^8 \cdot (f^{1/2}) = f^{3 + 1/2} g^5 h^8 = f^{3.5} g^5 h^8 \] Therefore, this expression simplifies to: \[ f^{7/2} g^5 h^8 \]
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Expression: \( h^8 |f^3 g^5| \sqrt{f} \)
Assuming \( f \geq 0 \), then \( |f| = f \), and the square root is: \[ \sqrt{f} = f^{1/2} \] Now substituting: \[ h^8 (f^3 g^5) (f^{1/2}) = h^8 f^{3 + 1/2} g^5 = h^8 f^{3.5} g^5 \] Therefore, this expression simplifies to: \[ h^8 f^{7/2} g^5 \]
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Expression: \( |f| \sqrt{f^3 g^5 h^8} \)
Again, assuming \( f \geq 0 \): \[ |f| = f \] Now simplify the square root: \[ \sqrt{f^3 g^5 h^8} = \sqrt{f^3} \sqrt{g^5} \sqrt{h^8} = f^{3/2} g^{5/2} h^4 \] Now substituting back: \[ f \cdot (f^{3/2} g^{5/2} h^4) = f^{1 + 3/2} g^{5/2} h^4 = f^{5/2} g^{5/2} h^4 \] Therefore, this expression simplifies to: \[ f^{5/2} g^{5/2} h^4 \]
Now, let's summarize the simplifications:
- \( f^2 g^3 h^4 \sqrt{f^3 g} = f^{9/2} g^{7/2} h^4 \)
- \( f^3 g^5 h^8 \sqrt{f} = f^{7/2} g^5 h^8 \)
- \( h^8 |f^3 g^5| \sqrt{f} = h^8 f^{7/2} g^5 \)
- \( |f| \sqrt{f^3 g^5 h^8} = f^{5/2} g^{5/2} h^4 \)