(a) Simplify : \(\frac{\frac{1}{2} of \frac{1}{4} \div \frac{1}{3}}{\frac{1}{6} - \frac{3}{4} + \frac{1}{2}}\). (b) Given that...
(a) Simplify : \(\frac{\frac{1}{2} of \frac{1}{4} \div \frac{1}{3}}{\frac{1}{6} - \frac{3}{4} + \frac{1}{2}}\).
(b) Given that \(\sqrt{x} = 10^{\bar{1}.6741}\), without using calculators, find the value of x.
Explanation
(a) \(\frac{\frac{1}{2} of \frac{1}{4} \div \frac{1}{3}}{\frac{1}{6} - \frac{3}{4} + \frac{1}{2}}\)
Numerator - \(\frac{1}{2} \times \frac{1}{4} \div \frac{1}{3} = (\frac{1}{2} \times \frac{1}{4}) \div \frac{1}{3}\)
\(\frac{1}{8} \times 3 = \frac{3}{8}\)
Denominator - \(\frac{1}{6} - \frac{3}{4} + \frac{1}{2} = \frac{1}{6} - (\frac{3}{4} - \frac{1}{2})\)
\(\frac{1}{6} - (\frac{3 - 2}{4}) = \frac{1}{6} - \frac{1}{4}\)
\(\frac{2 - 3}{12} = \frac{-1}{12}\)
\(\therefore \frac{\frac{1}{2} \times \frac{1}{4} \div \frac{1}{3}}{\frac{1}{6} - \frac{3}{4} + \frac{1}{2}} = \frac{3}{8} \div \frac{-1}{12}\)
\(\frac{3}{8} \times -12 = \frac{-9}{2}\)
= \(-4.5\)
(b) \(\sqrt{x} = 10^{\bar{1}.6741}}\)
Squaring both sides, we have
\(x = (10^{\bar{1}.6741}})^{2}\)
\(x = 10^{\bar{1}.3482}\) (checking antilogarithm)
\(x = 0.2229\)