We are given the following radical expression
![4\sqrt[]{28z}+\sqrt[]{63z}](https://img.qammunity.org/2023/formulas/mathematics/college/mp6ojng3t8tj5zmhgu54ovbdwvn00xhwv7.png)
Let us simplify the expression.
Re-write the radicals as
![4\sqrt[]{28z}+\sqrt[]{63z}=4\sqrt[]{4\cdot7z}+\sqrt[]{9\cdot7z}](https://img.qammunity.org/2023/formulas/mathematics/college/3z7ufzp79y6wn73817wqdqxhat612ktkb6.png)
Apply the product rule below
![\sqrt[]{a\cdot b}=\sqrt[]{a}\cdot\sqrt[]{b}](https://img.qammunity.org/2023/formulas/mathematics/college/hxyibywca68il82o4c5roud802ancab5oh.png)
So applying the above rule, the expression becomes
![4\sqrt[]{4\cdot7z}+\sqrt[]{9\cdot7z}=4\sqrt[]{4}\cdot\sqrt[]{7z}+\sqrt[]{9}\cdot\sqrt[]{7z}](https://img.qammunity.org/2023/formulas/mathematics/college/kvb09k9vl1zqz2j4u1ztxxkfh783tc8a7u.png)
We know that 4 and 9 are perfect squares so the expression becomes
![4\sqrt[]{4}\cdot\sqrt[]{7z}+\sqrt[]{9}\cdot\sqrt[]{7z}=4\cdot2\cdot\sqrt[]{7z}+3\cdot\sqrt[]{7z}=8\cdot\sqrt[]{7z}+3\cdot\sqrt[]{7z}](https://img.qammunity.org/2023/formulas/mathematics/college/vditctl673osejoopna1qhwdtzybt59bq9.png)
Finally, Combine the radicals
![8\cdot\sqrt[]{7z}+3\cdot\sqrt[]{7z}=(8+3)\sqrt[]{7z}_{}=11\sqrt[]{7z}](https://img.qammunity.org/2023/formulas/mathematics/college/35bnvrg6s8tww2k3ljpxonkg420v2gnxbh.png)
Therefore, the simplified expression is
![11\sqrt[]{7z}](https://img.qammunity.org/2023/formulas/mathematics/college/bqdrewvk2fel7r1t1m13nsdrsxgpn9p2pv.png)