Answer:
Option #1: [tex]6\frac{1}{2}[/tex]
Step-by-step explanation:
#1: Multiply [tex]\frac{\sqrt[3]{6}}{\sqrt[4]{6}}[/tex] and [tex]\frac{\sqrt[3]{6}}{\sqrt[4]{6}}[/tex]:
[tex]\frac{\sqrt[3]{6}}{\sqrt[4]{6}} * \frac{\sqrt[3]{6}}{\sqrt[4]{6}}[/tex]
#2: Combine and simplify the denominator:
- Multiply [tex]\frac{\sqrt[3]{6}}{\sqrt[4]{6}}[/tex] by [tex]\frac{\sqrt[3]{6}}{\sqrt[4]{6}}[/tex] = [tex]\frac{\sqrt[3]{6} \sqrt[4]{6}^{3} }{\sqrt[4]{6} \sqrt[4]{6}^3}[/tex]
- Raise [tex]\sqrt[4]{6}[/tex] to the power of 1
- Use the power rule [tex]a^{m} a^{n} =a^{m+n}[/tex] to combine exponents: [tex]\frac{\sqrt[3]{6} \sqrt[4]{6}^{3} }{\sqrt[4]{6}^{1+3}}[/tex]
- Add 1 and 3
- Rewrite [tex]\sqrt[4]{6}^4[/tex] as 6: [tex]\frac{\sqrt[3]{6} \sqrt[4]{6}^3}{6}[/tex]
#3: Simplify the numerator:
- Rewrite the expression using the least common index of 12: [tex]\frac{\sqrt[12]{6^4} \sqrt[12]{216^3}}{6}[/tex]
- Combine using the product rule for radicals: [tex]\frac{\sqrt[3]{6^4 *216^3}}{6}[/tex]
- Rewrite 216 as [tex]6^3[/tex]: [tex]\frac{\sqrt[3]{6^{4}*(6^{3})^{3}}}{6}[/tex]
- Multiply the exponents in [tex](6^{3})^3[/tex]: [tex]\frac{\sqrt[12]{6^{4}*6^{9}}}{6}[/tex]
- Use the power rule [tex]a^{m}a^{n}=a^{m+n}[/tex] to combine exponents and add [tex]4+9[/tex]:
[tex]\frac{\sqrt[12]{6^{13}}}{6}[/tex]
- Raise 6 to the power of 16: [tex]\frac{\sqrt[12]{13060694016}}{6}[/tex]
- Rewrite 13060694016 as [tex]6^{12}*6[/tex]: [tex]\frac{\sqrt[12]{6^{12}*6}}{6}[/tex]
- Pull terms out from under the radical: [tex]\frac{6\sqrt[12]{6}}{6}[/tex]
#4: Cancel the common factor of 6:
[tex]\frac{\sqrt[12]{6}}{6}=6\frac{1}{2}[/tex]
The correct simplified answer for [tex]\frac{\sqrt[3]{6}}{\sqrt[4]{6}}[/tex] is Option #1: [tex]6\frac{1}{2}[/tex].
