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Question

Question: How do you simplify \(\sqrt { - 144}\) ?...

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

Explanation

Solution

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 1\sqrt { - 1} as ii .

Complete step by step answer:
The given expression to simplify is 144\sqrt { - 144}
This term can also be written in the following way without changing its value,
144=144×1\sqrt { - 144} = \sqrt {144 \times - 1}
Applying the rule of surds that states a×b=a×b\sqrt {a \times b} = \sqrt a \times \sqrt b to the above expression we get
144×1\Rightarrow \sqrt {144} \times \sqrt { - 1} ...(i)...(i)
Now we know that (12)2=12×12=144{(12)^2} = 12 \times 12 = 144 therefore,
144=(144)1/2\sqrt {144} = {(144)^{1/2}}
Which on further simplification gives us
(144)1/2=(122)1/2{(144)^{1/2}} = {({12^2})^{1/2}}
Applying the law of exponents that states (am)n=am×n{({a^m})^n} = {a^{m \times n}} to the above expression,
(144)1/2=122×12{(144)^{1/2}} = {12^{2 \times \dfrac{1}{2}}}
On simplifying the powers of   12\;12 we get
144=12\Rightarrow \sqrt {144} = 12 ...(ii)...(ii)
Now on substituting equation (ii)(ii) in the equation (i)(i) we get
144×1=12×1\sqrt {144} \times \sqrt { - 1} = 12 \times \sqrt { - 1}
In complex and imaginary numbers we know that 1\sqrt { - 1} can be written as ii , therefore substituting the value of 1\sqrt { - 1} in the above expression we get
=12×i= 12 \times i
=12i= 12i

Hence, on simplifying 144\sqrt { - 144} we get 12i12i .

Additional information:
A complex number can be defined as a number which can be expressed in the form a+iba + ib where aa and bb are real numbers and ii represents the imaginary number and satisfies the equation i2=1{i^2} = - 1 . It also means that the value of ii is i=1i = \sqrt { - 1} . Since no real number satisfies the two given equations ii 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 (am)n=am×n{({a^m})^n} = {a^{m \times n}} and a×b=a×b\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 ii is i=1i = \sqrt { - 1} and not i=1i = 1 .