Simplify √(5-21i) and write the answer in the form `a+bi, where a,b ∈ R
let x = 5 - 21i
|x| = √(25 + 441) = √466
tanθ = -21/5
θ = 360° - 76.608° = 283.392° , since 5 - 21i is in quad IV on the Argand plane
x^(1/2) = √√466(cos (1/2)283.392°+ i sin (1/2)283.392) , using De Moivre
= √√466(cos 141.696 + i sin 141.696)
= 4.646(-.784735.. + i .61983..)
= -.3646 + 2.8798 i <------ primary square root
other root = √√466(cos (141.696 + 180°) + i sin (141.696 + 180°) )
leaving that calculation up to you
checking
( 3.646 + 2.8798 i)^2
= 13.2935 - 21i + 8.2935 i^2
= 13.2935 - 21i - 8.2935
= 5 - 21 i
To simplify the expression √(5-21i), we can use the properties of complex numbers.
Step 1: Write the expression in the form a+bi
The given expression is already in the form a+bi, with a = 5 and b = -21. It can be written as 5 - 21i.
Step 2: Find the magnitude of the expression
The magnitude (or modulus) of a complex number is given by |z| = √(a^2 + b^2), where z = a+bi.
For the expression 5 - 21i, the magnitude is |5 - 21i| = √(5^2 + (-21)^2) = √(25 + 441) = √466 = 2√116.
Step 3: Find the principal argument
The principal argument of a complex number in the form a+bi is given by θ = arctan(b/a).
For the expression 5 - 21i, the principal argument is θ = arctan((-21)/5).
Step 4: Simplify the square root
The expression √(5 - 21i) can be written as √(2√116 * e^(iθ)) since we know the magnitude and principal argument.
Using the properties of square roots and exponentials, √(2√116 * e^(iθ)) simplifies to √(2√116) * √(e^(iθ)).
Step 5: Evaluate the square root of the magnitude
√(2√116) can be further simplified as follows:
√(2√116) = √(2 * √(4 * 29)) = √(2 * 2 * √29) = 2 * √(√29) = 2 * (√(√29)).
Step 6: Evaluate √(e^(iθ))
To evaluate √(e^(iθ)), rewrite it in exponential form: e^(iθ) = cosθ + i*sinθ.
Taking the square root of e^(iθ) gives us two possibilities: ±√(cosθ + i*sinθ).
We need to evaluate both possibilities.
Step 7: Evaluate √(cosθ + i*sinθ) (first possibility)
Evaluating √(cosθ + i*sinθ) for the given expression 5 - 21i, we have:
√(cosθ + i*sinθ) = √(cos(arctan((-21)/5)) + i*sin(arctan((-21)/5))).
Using trigonometric identities, we can evaluate cos and sin of the arctan term.
Step 8: Evaluate √(cos(arctan((-21)/5)) + i*sin(arctan((-21)/5)))
Evaluate the real and imaginary parts separately:
Real part: cos(arctan((-21)/5))
Imaginary part: sin(arctan((-21)/5))
Step 9: Calculate cos(arctan((-21)/5))
Using a calculator or referring to trigonometric tables, calculate cos(arctan((-21)/5)).
Step 10: Calculate sin(arctan((-21)/5))
Using a calculator or referring to trigonometric tables, calculate sin(arctan((-21)/5)).
Step 11: Write the answer in the form a+bi
Using the values obtained in Steps 9 and 10, you can write the simplified expression in the form a+bi, where a and b are real numbers.