(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...
Question 1 Report
(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.
Answer Details
(a) Numerator: \(\tfrac{1}{2}\text{ of }\tfrac{1}{4} \div \tfrac{1}{3} = \tfrac{1}{8}\div\tfrac{1}{3} = \tfrac{1}{8}\times3 = \tfrac{3}{8}.\)
Denominator: \(\tfrac{1}{6} - \tfrac{3}{4} + \tfrac{1}{2} = \tfrac{2 - 9 + 6}{12} = -\tfrac{1}{12}.\)
\[\frac{\tfrac{3}{8}}{-\tfrac{1}{12}} = \tfrac{3}{8}\times(-12) = -\tfrac{9}{2} = -4\tfrac{1}{2}.\]
(b) The bar denotes a negative characteristic, so \(\bar{1}.6741 = -1 + 0.6741 = -0.3259\). Squaring \(\sqrt{x}\):
\[x = \left(10^{\bar{1}.6741}\right)^2 = 10^{\,2(-0.3259)} = 10^{-0.6518} = 10^{\bar{1}.3482}.\]
The antilog of \(0.3482\) is \(2.229\), and the characteristic \(\bar{1}\) places one zero after the point:
\[x = 0.2229 \approx 0.223.\]
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