Question

which expression is equivalent to (sqrt4{\frac{81}{16}a^{8}b^{12}c^{16}})?

• (\frac{3}{2}a^{2}b^{3}c^{4})
• (\frac{3}{2}a^{4}b^{8}c^{12})
• (\frac{9}{4}a^{2}b^{3}c^{4})
• (\frac{9}{4}a^{4}b^{6}c^{8})

Explanation:

Step1: Simplify the radical of the fraction

The fourth root of a fraction is the fraction of the fourth roots. So, $\sqrt[4]{\frac{81}{16}}=\frac{\sqrt[4]{81}}{\sqrt[4]{16}}$. Since $81 = 3^4$ and $16 = 2^4$, we have $\frac{\sqrt[4]{3^4}}{\sqrt[4]{2^4}}=\frac{3}{2}$.

Step2: Simplify the radical of $a^8$

Using the property $\sqrt[n]{x^m}=x^{\frac{m}{n}}$, for $\sqrt[4]{a^8}$, we get $a^{\frac{8}{4}} = a^2$.

Step3: Simplify the radical of $b^{12}$

For $\sqrt[4]{b^{12}}$, using the same property, we have $b^{\frac{12}{4}} = b^3$.

Step4: Simplify the radical of $c^{16}$

For $\sqrt[4]{c^{16}}$, we get $c^{\frac{16}{4}} = c^4$.

Step5: Combine all the simplified parts

Multiplying the results from each step together, we have $\frac{3}{2} \times a^2 \times b^3 \times c^4=\frac{3}{2}a^2b^3c^4$.

Answer:

$\frac{3}{2}a^2b^3c^4$ (the first option: $\frac{3}{2}a^2b^3c^4$)