Firstly, we can separate the factors with the variable x and the factors without it.
When we have a multiplication inside a root, we can separate them into two roots:
[tex]\sqrt[]{36x^8}\cdot\sqrt[4]{16x^{12}}=\sqrt[]{36}\sqrt[]{x^8}\cdot\sqrt[4]{16}\sqrt[4]{x^{12}}=\sqrt[]{36}\sqrt[4]{16}\cdot\sqrt[]{x^8}\sqrt[4]{x^{12}}[/tex]
For the factors without x, we can see that 36 is the same as 6 times 6 and 16 is the same as 2 * 2 * 2 * 2:
[tex]\sqrt[]{36}\sqrt[4]{16}\cdot\sqrt[]{x^8}\sqrt[4]{x^{12}}=\sqrt[]{6^2}\sqrt[4]{2^4}\cdot\sqrt[]{x^8}\sqrt[4]{x^{12}}[/tex]
Now, we can apply the following property for all roots:
[tex]\sqrt[c]{b^a}=b^{\frac{a}{c}}[/tex]
So, we have:
[tex]\sqrt[]{6^2}\sqrt[4]{2^4}\cdot\sqrt[]{x^8}\sqrt[4]{x^{12}}=6^{\frac{2}{2}}2^{\frac{4}{4}}\cdot x^{\frac{8}{2}}x^{\frac{12}{4}}=6\cdot2\cdot x^4x^3=12\cdot x^4x^3[/tex]
Now, we can use the property:
[tex]b^ab^c=b^{a+c}[/tex]
So:
[tex]12x^4x^3=12x^{4+3}=12x^7[/tex]
So, the simplest form is:
[tex]12x^7[/tex]
