Question

How do you simplify $\sqrt { - 144}$ ?

Answer 1

To solve such questions that contain a negative sign under the root, we require the basic knowledge of complex or imaginary numbers. The given expression can be written as an imaginary number by substituting the value of $\sqrt { - 1}$ as $i$ .

Complete step by step answer:
The given expression to simplify is $\sqrt { - 144}$
This term can also be written in the following way without changing its value,
$\sqrt { - 144} = \sqrt {144 \times - 1}$
Applying the rule of surds that states $\sqrt {a \times b} = \sqrt a \times \sqrt b$ to the above expression we get
$\Rightarrow \sqrt {144} \times \sqrt { - 1}$ $...(i)$
Now we know that ${(12)^2} = 12 \times 12 = 144$ therefore,
$\sqrt {144} = {(144)^{1/2}}$
Which on further simplification gives us
${(144)^{1/2}} = {({12^2})^{1/2}}$
Applying the law of exponents that states ${({a^m})^n} = {a^{m \times n}}$ to the above expression,
${(144)^{1/2}} = {12^{2 \times \dfrac{1}{2}}}$
On simplifying the powers of $\;12$ we get
$\Rightarrow \sqrt {144} = 12$ $...(ii)$
Now on substituting equation $(ii)$ in the equation $(i)$ we get
$\sqrt {144} \times \sqrt { - 1} = 12 \times \sqrt { - 1}$
In complex and imaginary numbers we know that $\sqrt { - 1}$ can be written as $i$ , therefore substituting the value of $\sqrt { - 1}$ in the above expression we get
$= 12 \times i$
$= 12i$

Hence, on simplifying $\sqrt { - 144}$ we get $12i$ .

Additional information:
A complex number can be defined as a number which can be expressed in the form $a + ib$ where $a$ and $b$ are real numbers and $i$ represents the imaginary number and satisfies the equation ${i^2} = - 1$ . It also means that the value of $i$ is $i = \sqrt { - 1}$ . Since no real number satisfies the two given equations $i$ is called an imaginary number. Complex numbers cannot be marked on the number line.

Note: While solving these types of questions it always proves extremely helpful if students remember the fundamental rules of surds and exponents. Some of the rules such as ${({a^m})^n} = {a^{m \times n}}$ and $\sqrt {a \times b} = \sqrt a \times \sqrt b$ are used a lot of times and help to simplify the question to a great extent. Also, keep in mind that the value of $i$ is $i = \sqrt { - 1}$ and not $i = 1$ .