To simplify the expression \(\sqrt{75yz^2}\), we need to break down the number inside the square root into its prime factors and identify any perfect squares.
First, we can factor \(75\):
\[ 75 = 25 \cdot 3 = 5^2 \cdot 3 \]
So, we can rewrite the expression:
\[ \sqrt{75yz^2} = \sqrt{25 \cdot 3 \cdot y \cdot z^2} \]
Next, we can separate the square root into the products of the square roots of its factors:
\[ \sqrt{75yz^2} = \sqrt{25} \cdot \sqrt{3} \cdot \sqrt{y} \cdot \sqrt{z^2} \]
Now we can simplify the square roots of the perfect squares:
\[ \sqrt{25} = 5 \quad \text{and} \quad \sqrt{z^2} = z \]
Substituting these back into our expression, we get:
\[ \sqrt{75yz^2} = 5 \cdot z \cdot \sqrt{3} \cdot \sqrt{y} \]
This simplifies to:
\[ 5z\sqrt{3y} \]
Thus, the simplified expression is:
\[ \sqrt{75yz^2} = 5z\sqrt{3y} \]
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