To simplify the expression [tex]\(\sqrt{252 x^8 y^{19}}\)[/tex], follow these steps:
1. Break down the expression under the square root:
[tex]\[
\sqrt{252 x^8 y^{19}}
\][/tex]
2. Factorize the constant and the variables separately inside the square root:
- For the constant [tex]\(252\)[/tex]:
[tex]\[
252 = 2^2 \times 3^2 \times 7
\][/tex]
So,
[tex]\[
\sqrt{252} = \sqrt{2^2 \times 3^2 \times 7} = \sqrt{(2 \times 3)^2 \times 7} = \sqrt{36 \times 7} = 6\sqrt{7}
\][/tex]
- For the variable [tex]\(x^8\)[/tex]:
[tex]\[
\sqrt{x^8} = x^{8/2} = x^4
\][/tex]
- For the variable [tex]\(y^{19}\)[/tex]:
[tex]\[
y^{19} = y^{18} \cdot y = (y^9)^2 \cdot y
\][/tex]
Which means,
[tex]\[
\sqrt{y^{19}} = \sqrt{(y^9)^2 \cdot y} = y^9 \sqrt{y}
\][/tex]
3. Combine these simplified parts together:
- The constant part [tex]\(6\sqrt{7}\)[/tex]
- The simplified variable [tex]\(x^4\)[/tex]
- The simplified variable [tex]\(y^9\sqrt{y}\)[/tex]
Therefore, the overall simplified expression is:
[tex]\[
6 \sqrt{7} \cdot x^4 \cdot y^9 \cdot \sqrt{y}
\][/tex]
4. Combine the results into a single expression:
[tex]\[
\boxed{6 \sqrt{7} x^4 y^9 \sqrt{y}}
\][/tex]
