Question

rewrite \\(\sqrt{\sqrt{196}}\\) in exponential form.\\(\sqrt{\sqrt{196}}=\square\\)\\(\text{hide math input}\\)\\(\text{basic}\quad\text{symbols}\quad\text{greek}\quad\text{other}\\)\\(\text{question 8}\\)\\(\text{simplify: } \left(\frac{1}{4}a^{\frac{2}{3}}b^{\frac{1}{4}}\
ight)\left(\frac{8}{5}a^{\frac{1}{3}}b^{\frac{3}{4}}\
ight)\\)\\(\circ\\ \frac{2}{20}a^{\frac{2}{3}}b^{\frac{1}{4}}\\)\\(\circ\\ \frac{12}{20}a^{\frac{8}{9}}b^{\frac{3}{16}}\\)\\(\circ\\ \frac{2}{5}a^{\frac{2}{3}}b^{\frac{1}{4}}\\)\\(\circ\\ \frac{2}{5}a^{\frac{8}{9}}b^{\frac{3}{16}}\\)

Explanation:

Step1: Rewrite root as exponent

Recall $\sqrt{x}=x^{\frac{1}{2}}$, so $\sqrt{108}=108^{\frac{1}{2}}$

Step2: Multiply coefficients

Multiply $\frac{4}{5} \times \frac{3}{5} = \frac{12}{25}$

Step3: Combine $a$ terms

Use $x^m \times x^n = x^{m+n}$: $a^{\frac{1}{2}} \times a^{\frac{2}{3}} = a^{\frac{3}{6}+\frac{4}{6}} = a^{\frac{7}{6}}$

Step4: Combine $b$ terms

Use $x^m \times x^n = x^{m+n}$: $b^{\frac{1}{4}} \times b^{\frac{3}{2}} = b^{\frac{1}{4}+\frac{6}{4}} = b^{\frac{7}{4}}$

Step5: Combine all parts

$\frac{12}{25}a^{\frac{7}{6}}b^{\frac{7}{4}}$

Answer:

1. $108^{\frac{1}{2}}$
2. B. $\frac{12}{25}a^{\frac{7}{6}}b^{\frac{7}{4}}$