Question

Solve the given exponent:

$\sqrt[4]{412}\times \sqrt[7]{76}$

• A. $2^{\frac{9}{14}}\times 3^{\frac{11}{28}}$
• B. $3^{\frac{9}{14}}\times 2^{\frac{11}{28}}$
• C. $2^{\frac{1}{14}}\times 3^{\frac{1}{28}}$
• D. $3^{\frac{1}{14}}\times 2^{\frac{1}{28}}$

Answer 1

The correct option is A $2^{\frac{9}{14}}\times 3^{\frac{11}{28}}$

$\sqrt[4]{412}\times \sqrt[7]{76}$

$=(12{)}^{\frac{1}{4}}\times (6{)}^{\frac{1}{7}}$

$=(2\times 2\times 3{)}^{\frac{1}{4}}\times (2\times 3{)}^{\frac{1}{7}}$

$=(2^2\times 3{)}^{\frac{1}{4}}\times (2\times 3{)}^{\frac{1}{7}}$

$=2^{2\times \left(\frac{1}{4}\right)}\times 3^{\frac{1}{4}}\times 2^{\frac{1}{7}}\times 3^{\frac{1}{7}}$

$=2^{\frac{1}{2}}\times 3^{\frac{1}{4}}\times 2^{\frac{1}{7}}\times 3^{\frac{1}{7}}$

$=2^{(\frac{1}{2}+\frac{1}{7})}\times 3^{(\frac{1}{4}+\frac{1}{7})}$-----If base is same, then their powers can be added, by product law.

$=2^{\frac{9}{14}}\times 3^{\frac{11}{28}}$

Option A.