simplify √45xy^3z^2

Answer 1

To simplify the expression \( \sqrt{45xy^3z^2} \), we can break it down as follows:

1. Factor \( 45 \) into its prime factors: \[ 45 = 9 \times 5 = 3^2 \times 5 \] Thus, we have: \[ \sqrt{45} = \sqrt{3^2 \times 5} = \sqrt{3^2} \times \sqrt{5} = 3\sqrt{5} \]

2. Now, let's simplify \( \sqrt{xy^3z^2} \):

   • For \( y^3 \): \( \sqrt{y^3} = \sqrt{y^2 \cdot y} = y \sqrt{y} \)
   • For \( z^2 \): \( \sqrt{z^2} = z \)

So, we have: \[ \sqrt{xy^3z^2} = \sqrt{x} \cdot \sqrt{y^3} \cdot \sqrt{z^2} = \sqrt{x} \cdot (y \sqrt{y}) \cdot z = y z \sqrt{xy} \]

3. Now combine everything: \[ \sqrt{45xy^3z^2} = \sqrt{45} \cdot \sqrt{xy^3z^2} = 3\sqrt{5} \cdot (y z \sqrt{xy}) \]

4. Finally, we can write it all together: \[ \sqrt{45xy^3z^2} = 3yz\sqrt{5xy} \]

Thus, the simplified form is: \[ \boxed{3yz\sqrt{5xy}} \]