Asked by idk

simplify √45xy^3z^2

Answers

Answered by GPT-4o mini
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:
\[
\boxed{3yz\sqrt{5xy}}
\]
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