Question

Simplify: \[4\sqrt { - 4} + 5\sqrt { - 9} - 3\sqrt { - 16} \]

Answer 1

Hint: First we are going to separate the terms inside the square root i.e., $\sqrt { - 4} $ as $\sqrt 4 \times \sqrt { - 1} $. As we know that the square root of $ - 1$ is i, by applying this is the given equation we can find the value. And the answer will be in terms of a complex number.

Complete step-by-step solution:
The given equation in the question is \[4\sqrt { - 4} + 5\sqrt { - 9} - 3\sqrt { - 16} \]
In the above equation, $\sqrt { - 4} $ can be written as $\sqrt 4 \times \sqrt { - 1} $, $\sqrt { - 9} $ can be written as $\sqrt 9 \times \sqrt { - 1} $ , and $\sqrt { - 16} $ can be written as $\sqrt {16} \times \sqrt { - 1} $ .
By applying this in the given equation, we get
$4(\sqrt 4 \times \sqrt { - 1} ) + 5(\sqrt 9 \times \sqrt { - 1} ) - 3(\sqrt {16} \times \sqrt { - 1} )$
As we know that the square root of -1 i.e., $\sqrt { - 1} $ is i, applying this on the above equation in the place square root of -1, we get
$4(\sqrt 4 \times \sqrt { - 1} ) + 5(\sqrt 9 \times \sqrt { - 1} ) - 3(\sqrt {16} \times \sqrt { - 1} )$ = $4(\sqrt 4 \times i) + 5(\sqrt 9 \times i) - 3(\sqrt {16} \times i)$
We know that the value of the square root of $4$ is \[2\], the value of the square root of $9$ is $3$, and the value of the square root of $16$ is $4$. By applying these values in the above equation we get,
$4(\sqrt 4 \times i) + 5(\sqrt 9 \times i) - 3(\sqrt {16} \times i)$= $4(2 \times i) + 5(3 \times i) - 3(4 \times i)$
First multiplying the terms and then adding those terms, we get
$4(2 \times i) + 5(3 \times i) - 3(4 \times i)$ = $8i + 15i - 12i = 11i$
Which is the required value for the given equation.

Note:
> Since the roots are in negative terms, we got the answer in complex numbers. The answer will be incorrect if we took the values as positive since they are in roots.
> If the roots are in positive terms, (for example), \[4\sqrt 4 + 5\sqrt 9 - 3\sqrt {16} \] , in this case, we can directly substitute the values of square root terms and we can find the answer directly. We don’t need to separate the terms inside the square root as we did in the above. And the answer will be in real numbers.