Question

Simplify the given expression $\sqrt{\dfrac{-45}{16}}+\sqrt{\dfrac{-20}{256}}$

Answer 1

Hint: Here, we use the concept of imaginary numbers, i.e. $\sqrt{-1}=i$. The unit imaginary number, i, equals the square root of minus 1. Imaginary numbers are not "imaginary", they really exist

Complete step-by-step answer:
We have to simplify $\sqrt{\dfrac{-45}{16}}+\sqrt{\dfrac{-20}{256}}$.
This expression can be written as:
$\sqrt{\dfrac{-45}{16}}+\sqrt{\dfrac{-20}{256}}=\sqrt{\dfrac{-45}{16}}+\sqrt{\dfrac{-5\times 4}{64\times 4}}$
Simplifying, we get,
$\sqrt{\dfrac{-45}{16}}+\sqrt{\dfrac{-20}{256}}=\sqrt{\dfrac{-45}{16}}+\sqrt{\dfrac{-5}{64}}$
Writing the terms as product of two numbers, we get,
$\sqrt{\dfrac{-45}{16}}+\sqrt{\dfrac{-20}{256}}=\sqrt{\dfrac{-5\times 9}{4 \times 4}}+\sqrt{\dfrac{-5}{8 \times 8}}$
Taking square root of the numbers, we get,
$\sqrt{\dfrac{-45}{16}}+\sqrt{\dfrac{-20}{256}}=\dfrac{3}{4}\sqrt{-5}+\dfrac{1}{8}\sqrt{-5}$
Adding the terms, we get,
$\sqrt{\dfrac{-45}{16}}+\sqrt{\dfrac{-20}{256}}=\left(\dfrac{3}{4}+\dfrac{1}{8}\right)\sqrt{-5}=\dfrac{7}{8}\sqrt{-5}$
Now, as $\sqrt{-1}=i$, thus,
$\sqrt{\dfrac{-45}{16}}+\sqrt{\dfrac{-20}{256}}=\dfrac{7\sqrt{5}i}{8}$
Hence, the simplified form of $\sqrt{\dfrac{-45}{16}}+\sqrt{\dfrac{-20}{256}}$ is $\dfrac{7\sqrt{5}i}{8}$.

Note: Imaginary numbers are the numbers that are not real. If we square imaginary numbers, we get negative results. Imaginary numbers can be expressed in the form of real numbers by multiplying them by i.