which expression is equivalent to $sqrt4{\frac{81}{16}a^{8}b^{12}c^{16}}$?\n$\frac{3}{2}a^{2}|b^{3}|c^{4}$\n$…

which expression is equivalent to $sqrt4{\frac{81}{16}a^{8}b^{12}c^{16}}$?\n$\frac{3}{2}a^{2}|b^{3}|c^{4}$\n$\frac{3}{2}a^{4}b^{8}c^{12}$\n$\frac{9}{4}a^{2}|b^{3}|c^{4}$\n$\frac{9}{4}a^{4}b^{6}c^{8}$

which expression is equivalent to $sqrt4{\frac{81}{16}a^{8}b^{12}c^{16}}$?\n$\frac{3}{2}a^{2}|b^{3}|c^{4}$\n$\frac{3}{2}a^{4}b^{8}c^{12}$\n$\frac{9}{4}a^{2}|b^{3}|c^{4}$\n$\frac{9}{4}a^{4}b^{6}c^{8}$

Answer

Explanation:

Step1: Apply root - fraction rule

$\sqrt[4]{\frac{81}{16}a^{8}b^{12}c^{16}}=\sqrt[4]{\frac{81}{16}}\cdot\sqrt[4]{a^{8}}\cdot\sqrt[4]{b^{12}}\cdot\sqrt[4]{c^{16}}$

Step2: Calculate $\sqrt[4]{\frac{81}{16}}$

$\sqrt[4]{\frac{81}{16}}=\frac{\sqrt[4]{81}}{\sqrt[4]{16}}=\frac{3}{2}$

Step3: Calculate $\sqrt[4]{a^{8}}$

$\sqrt[4]{a^{8}}=a^{\frac{8}{4}} = a^{2}$

Step4: Calculate $\sqrt[4]{b^{12}}$

$\sqrt[4]{b^{12}}=\vert b^{3}\vert$ (since for even - root $\sqrt[n]{x^{n}}=\vert x\vert$ when $n$ is even)

Step5: Calculate $\sqrt[4]{c^{16}}$

$\sqrt[4]{c^{16}}=c^{\frac{16}{4}}=c^{4}$

Step6: Combine the results

$\frac{3}{2}a^{2}\vert b^{3}\vert c^{4}$

Answer:

$\frac{3}{2}a^{2}\vert b^{3}\vert c^{4}$