Question

Find the product of additive inverse of $ \dfrac{-2}{7}\times \dfrac{4}{5} $ and multiplicative inverse of $ \dfrac{13}{21} $ . Write your answer in the simplest form.

Answer 1

Hint: We start solving this problem by finding the additive inverse of the obtained number after multiplying given numbers. Then we find the multiplicative inverse of the other given number and then multiply obtained results to obtain the final answer.

Complete step-by-step answer:
Additive inverse of a number is a real number which sums up to the given number and gives zero.
Let us consider the definition of additive inverse.
Let a and b be two real numbers. Then we say b is the additive inverse of a, if
 $ a+b=0 $ .
Let us consider the given number $ \dfrac{-2}{7}\times \dfrac{4}{5} $ as a.
Now, we need to find the additive inverse of a that is b.
Then, by the definition above, we get
\[\begin{align}
  & \Rightarrow \left( \dfrac{-2}{7}\times \dfrac{4}{5} \right)+b=0 \\
 & \Rightarrow \left( \dfrac{-8}{35} \right)+b=0 \\
 & \Rightarrow b=-\left( \dfrac{-8}{35} \right) \\
 & \Rightarrow b=\dfrac{8}{35} \\
\end{align}\]
So, the additive inverse of $ \dfrac{-2}{7}\times \dfrac{4}{5} $ is \[\dfrac{8}{35}\].
Now, let us consider the definition of Multiplicative Inverse.
Multiplicative inverse of a number is nothing but the reciprocal of the number, i.e., the multiplicative inverse of x is $ \dfrac{1}{x} $ .
We need to find the multiplicative inverse of $ \dfrac{13}{21} $ .
So, from the definition above, we get
\[\begin{align}
  & \Rightarrow \dfrac{1}{\dfrac{13}{21}} \\
 & \Rightarrow \dfrac{21}{13} \\
\end{align}\]
Now, we need to multiply the both obtained results to get the required result.
So, multiplying them we get
\[\Rightarrow \dfrac{8}{35}\times \dfrac{21}{13}\]
As both 21, 35 are divisible by 7 we reduce them. Then we get
\[\begin{align}
  & \Rightarrow \dfrac{8}{5}\times \dfrac{3}{13} \\
 & \Rightarrow \dfrac{24}{65} \\
\end{align}\]
Therefore, the product of additive inverse $ \dfrac{-2}{7}\times \dfrac{4}{5} $ of and multiplicative inverse of $ \dfrac{13}{21} $ is \[\dfrac{24}{65}\].
Hence the answer is \[\dfrac{24}{65}\].

Note: While finding the additive inverse of $ \dfrac{-2}{7}\times \dfrac{4}{5} $ , there is a chance of making mistake by finding additive inverse for both the numbers and multiply them like
Additive inverse of $ \dfrac{-2}{7} $ is $ \dfrac{-2}{7}+x=0\Rightarrow x=-\left( \dfrac{-2}{7} \right)=\dfrac{2}{7} $ .
Additive inverse of $ \dfrac{4}{5} $ is $ \dfrac{4}{5}+x=0\Rightarrow x=-\left( \dfrac{4}{5} \right)=-\dfrac{4}{5} $ .