Question

simplify the radical. assume that all variables represent positive real numbers. \\(\sqrt4{a^8b^{12}}\\) select the correct choice below and, if necessary, fill in the answer box to complete your ch \\(\bigcirc\\) a. \\(\sqrt4{a^8b^{12}} = \square\\) \\(\bigcirc\\) b. the root is not a real number.

Explanation:

Step1: Apply the nth root property

For a radical \(\sqrt[n]{x^m}\), it can be simplified as \(x^{\frac{m}{n}}\) when \(x\) is non - negative (here variables are positive real numbers). For the fourth root of \(a^{8}b^{12}\), we can use the property \(\sqrt[n]{ab}=\sqrt[n]{a}\cdot\sqrt[n]{b}\) (product rule of radicals) and \(\sqrt[n]{x^m}=x^{\frac{m}{n}}\).

First, simplify \(\sqrt[4]{a^{8}}\): Using the formula \(\sqrt[n]{x^m}=x^{\frac{m}{n}}\), here \(n = 4\) and \(m=8\), so \(\sqrt[4]{a^{8}}=a^{\frac{8}{4}}=a^{2}\).

Step2: Simplify \(\sqrt[4]{b^{12}}\)

Using the same formula \(\sqrt[n]{x^m}=x^{\frac{m}{n}}\) with \(n = 4\) and \(m = 12\), we get \(\sqrt[4]{b^{12}}=b^{\frac{12}{4}}=b^{3}\).

Step3: Combine the results

Since \(\sqrt[4]{a^{8}b^{12}}=\sqrt[4]{a^{8}}\cdot\sqrt[4]{b^{12}}\), substituting the simplified forms of \(\sqrt[4]{a^{8}}\) and \(\sqrt[4]{b^{12}}\), we have \(\sqrt[4]{a^{8}b^{12}}=a^{2}b^{3}\).

Answer:

A. \(\sqrt[4]{a^{8}b^{12}}=\boldsymbol{a^{2}b^{3}}\)