Question

The value of $\left(7^{-1}-8^{-1}\right)^{-1}-\left(3^{-1}-4^{-1}\right) ^{-1}$ is
(a)44
(b)56
(c)68
(d)12

Answer 1

Hint: We need to solve a mathematical expression here. We need to be really aware while opening the brackets and doing the inverse operations. For that we will simply do the operations of each bracket separately and later on put them both together so that we make no mistakes at all. We will also use some formulae of fraction subtraction. Also, some formulae involving the negative power and inverses in the fractions will also be used.

Complete step-by-step answer:
We are given $\left(7^{-1}-8^{-1}\right)^{-1}-\left(3^{-1}-4^{-1}\right) ^{-1}$. We look at the first bracket:
$\left(7^{-1}-8^{-1}\right)^{-1}$
This can be written as follows:
$\left(\dfrac{1}{7}-\dfrac{1}{8}\right)^{-1}$
We apply the following formula to solve this fraction:
$\dfrac{1}{a}-\dfrac{1}{b}=\dfrac{b-a}{a\times b}$
So, we get the following:
$\left(\dfrac{1}{7}-\dfrac{1}{8}\right) ^{-1}=\left(\dfrac{8-7}{8\times 7}\right)^{-1}$
$\implies \left(\dfrac{1}{7}-\dfrac{1}{8}\right) ^{-1}=\left(\dfrac{1}{56}\right)^{-1}=\left(56^{-1}\right)^{-1}=56$
We have used the following property here:
$\left(a^{-1}\right)^{-1}=a$
We have now solved the first bracket we now move to the next bracket
$\left(3^{-1}-4^{-1}\right) ^{-1}$
We use the same set of formulae here as well:
$\left(3^{-1}-4^{-1}\right) ^{-1}= \left(\dfrac{1}{3}-\dfrac{1}{4}\right) ^{-1}=\left(\dfrac{4-3}{4\times 3}\right)^{-1}$
Simplifying this further we obtain:
$\left(3^{-1}-4^{-1}\right) ^{-1}= \left(\dfrac{1}{12}\right) ^{-1}=12$
The same formula for removing the inverse has been used here as well.
We now combine the two brackets and produce the resultant output:
$\left(7^{-1}-8^{-1}\right)^{-1}-\left(3^{-1}-4^{-1}\right) ^{-1}=\left(56\right)- \left(12\right)=44$
So, we get 44 as the answer.
Hence, the correct option is $\left(a\right)$
So, the correct answer is “Option A”.
Note: While opening the bracket involving the negative sign, make sure that you put the negative sign on both the terms inside the bracket. Moreover, while putting the inverse signs make sure that you first resolve the quantity in the bracket and after that you put inverse on the whole expression. Do not put inverse separately on each of the quantity because that would lead to an invalid result. Always solve the brackets individually first and then merge them.