JAMB Mathematics 2018 Paper

Concept:When dividing powers that have the same base, subtract the exponents.Explanation:The expression to simplify is $\frac{25^{\frac{2}{3}} \div 25^{\frac{1}{6}}}{\left(\frac{1}{5}\right)^{\frac{7}{6}} \div \left(\frac{1}{5}\right)^{\frac{1}{6}}}$.Begin with the numerator.Since both terms have the same base $25$, subtract the exponents:$25^{\frac{2}{3}} \div 25^{\frac{1}{6}} = 25^{\frac{2}{3} - \frac{1}{6}}$.Write the fractions with a common denominator:$\frac{2}{3} - \frac{1}{6} = \frac{4}{6} - \frac{1}{6} = \frac{3}{6} = \frac{1}{2}$.Therefore, the numerator is $25^{\frac{1}{2}} = \sqrt{25} = 5$.Now consider the denominator.The base is $\frac{1}{5}$, so subtract the exponents too:$\left(\frac{1}{5}\right)^{\frac{7}{6}} \div \left(\frac{1}{5}\right)^{\frac{1}{6}} = \left(\frac{1}{5}\right)^{\frac{7}{6} - \frac{1}{6}}$.This simplifies to $\left(\frac{1}{5}\right)^{\frac{6}{6}} = \left(\frac{1}{5}\right)^1 = \frac{1}{5}$.The whole fraction now reads $5 \div \frac{1}{5}$.Dividing by a fraction means multiplying by its reciprocal:$5 \times 5 = 25$.Answer:The simplified value is $25$, corresponding to option A.