As given by the question
There are given that the expression
[tex]6\sqrt[]{75x^9y^{12}z}[/tex]
Now,
Simplify the above expression
First, apply the radical rule;
[tex]\sqrt[]{ab}=\sqrt[]{a}\sqrt[]{b}[/tex]
Then,
From the given expression
[tex]6\sqrt[]{75x^9y^{12}z}=6\sqrt[]{75}\sqrt[]{x^9y^{12}z}^{}[/tex]
Then,
[tex]6\sqrt[]{75}\sqrt[]{x^9y^{12}z}^{}=6\sqrt[]{75}\sqrt[]{x^9}\sqrt[]{y^{12}z}[/tex]
Now,
[tex]6\sqrt[]{75}\sqrt[]{x^9}\sqrt[]{y^{12}z}=6\times5\sqrt[]{3}\sqrt[]{x^9}\sqrt[]{y^{12}}\text{ }\sqrt[]{z}[/tex]
Then,
Simplify the above equation again
So,
[tex]\begin{gathered} 6\times5\sqrt[]{3}\sqrt[]{x^9}\sqrt[]{y^{12}}\text{ }\sqrt[]{z}=6\times5\sqrt[]{3\times}^{}\sqrt[]{x^8\times x}\times\sqrt[]{y^{12}}\sqrt[]{z} \\ =6\times5\sqrt[]{3\times}x^4\sqrt[]{x}\times\sqrt[]{y^{12}}\sqrt[]{z} \end{gathered}[/tex]
Then,
Simplify the y terms in the above result
So,
[tex]\begin{gathered} 6\times5\sqrt[]{3\times}x^4\sqrt[]{x}\times\sqrt[]{y^{12}}\sqrt[]{z}=6\times5\sqrt[]{3\times}x^4\sqrt[]{x}\times y^6\sqrt[]{z} \\ \end{gathered}[/tex]
Now,
Multiply 6 with 5 and written in the standard form
So,
[tex]6\times5\sqrt[]{3\times}x^4\sqrt[]{x}\times y^6\sqrt[]{z}=30x^4y^6\sqrt[]{x}\sqrt[]{z}[/tex]
Hence, the value of the given expression is shown below:
[tex]30\sqrt[]{3}x^4y^6\sqrt[]{x}\sqrt[]{z}[/tex]
