Question
Question: How do you simplify \(\sqrt { - 144}\) ?...
How do you simplify −144 ?
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 as i .
Complete step by step answer:
The given expression to simplify is −144
This term can also be written in the following way without changing its value,
−144=144×−1
Applying the rule of surds that states a×b=a×b to the above expression we get
⇒144×−1 ...(i)
Now we know that (12)2=12×12=144 therefore,
144=(144)1/2
Which on further simplification gives us
(144)1/2=(122)1/2
Applying the law of exponents that states (am)n=am×n to the above expression,
(144)1/2=122×21
On simplifying the powers of 12 we get
⇒144=12 ...(ii)
Now on substituting equation (ii) in the equation (i) we get
144×−1=12×−1
In complex and imaginary numbers we know that −1 can be written as i , therefore substituting the value of −1 in the above expression we get
=12×i
=12i
Hence, on simplifying −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 i2=−1 . It also means that the value of i is i=−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 (am)n=am×n and a×b=a×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=−1 and not i=1 .