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Find the values of each of the following:
(i) \( \left(\frac{1}{2}\right)^{-1}+\left(\frac{1}{3}\right)^{-1}+\left(\frac{1}{4}\right)^{-1} \)
(ii) \( \left(\frac{1}{2}\right)^{-2}+\left(\frac{1}{3}\right)^{-2}+\left(\frac{1}{4}\right)^{-2} \)
(iii) \( \left(2^{-1} \times 4^{-1}\right) \div 2^{-2} \)
(iv) \( \left(5^{-1} \times 2^{-1}\right) \div 6^{-1} \)
To do:
We have to find the values of each of the given expressions.
Solution:
We know that,
$a^{-m}=\frac{1}{a^{m}}$
Therefore,
(i) $(\frac{1}{2})^{-1}+(\frac{1}{3})^{-1}+(\frac{1}{4})^{-1}=(2)^{1}+(3)^{1}+(4)^{1}$
$=2+3+4$
$=9$
(ii) $(\frac{1}{2})^{-2}+(\frac{1}{3})^{-2}+(\frac{1}{4})^{-2}=(2)^{2}+(3)^{2}+(4)^{2}$
$=4+9+16$
$=29$
(iii) $(2^{-1} \times 4^{-1}) \div 2^{-2}=(\frac{1}{2} \times \frac{1}{4}) \div \frac{1}{2^{2}}$
$=\frac{1}{8} \div \frac{1}{4}$
$=\frac{1}{8} \times \frac{4}{1}$
$=\frac{1}{2}$
(iv) $(5^{-1} \times 2^{-1}) \div 6^{-1}=(\frac{1}{5} \times \frac{1}{2}) \div \frac{1}{6}$
$=\frac{1}{10} \times \frac{6}{1}$
$=\frac{6}{10}$
$=\frac{3}{5}$