Given that real parts of √(5+12 i) and √(5-12 i) are negative, the number (√(5+12 i+√(5-12 i)))/

Given that real parts of √(5+12 i) and √(5-12 i) are...

Given that real parts of $\sqrt{5+12 \mathrm{i}}$ and $\sqrt{5-12 \mathrm{i}}$ are negative, the number $\frac{\sqrt{5+12 \mathrm{i}+\sqrt{5-12 \mathrm{i}}}}{\sqrt{5+12 \mathrm{i}}-\sqrt{5-12 \mathrm{i}}}$ reduces to (a) $\frac{3}{2} \mathrm{i}$ (b) $-\frac{3}{2} \mathrm{i}$ (c) $-3+\frac{2}{5} i$ (d) None of these

Given that real parts of $\sqrt{5+12 \mathrm{i}}$ and $\sqrt{5-12 \mathrm{i}}$ are negative, the number $\frac{\sqrt{5+12 \mathrm{i}+\sqrt{5-12 \mathrm{i}}}}{\sqrt{5+12 \mathrm{i}}-\sqrt{5-12 \mathrm{i}}}$ reduces to
(a) $\frac{3}{2} \mathrm{i}$
(b) $-\frac{3}{2} \mathrm{i}$
(c) $-3+\frac{2}{5} i$
(d) None of these

Key Concepts

Algebraic Simplification in the Complex Domain

Simplifying expressions involving complex numbers, radicals, and fractions often requires careful rearrangement and the use of identities specific to complex arithmetic. This includes recognizing properties like the relationship between complex conjugates and how the choice of branch affects further simplification.

Branch Selection in Complex Functions

When evaluating multi-valued functions like the square root over the complex numbers, it is essential to choose a specific branch or determination of the function to avoid ambiguity. The branch selection, such as prescribing the sign of the real part, ensures that the resulting function behaves consistently in further algebraic manipulations.

Complex Numbers

Complex numbers extend the real numbers by incorporating the imaginary unit i, where i^2 = -1. They allow for a complete solution of polynomial equations and are represented in the form a + bi, where a and b are real numbers. The arithmetic of complex numbers, including addition, multiplication, and finding roots, is fundamental in advanced algebra and calculus.

Square Roots of Complex Numbers

Finding the square root of a complex number involves determining another complex number which, when squared, returns the original number. Because every non-zero complex number has two square roots, careful selection of the 'branch' or specific value, such as considering the real part’s sign, is necessary to maintain consistency in computations.

Transcript

00:01 In this question it is given that the real parts of under root 5 plus 12 iota and under root 5 minus 12 iota are negative.

00:26 Part are negative.

00:33 Then we need to find the value of under root 5 plus 12 iota plus under root 5 minus 12 iota plus under root 5 minus 12 iota upon under root 5 plus 12 iota minus under root 5 minus 12 iota let's see how to solve this question consider that under root 5 plus 12 iota is equal to a plus iota b now square the both sides square b both sides we get 5 plus 12 iota is equal to a square plus iota b square plus 2 iota a b since iota square is equal to minus 1 therefore we can write 5 plus 12 iota is equal to a a square minus b square plus two iota a b .b.

02:22 Now equate the real part, we get 5 is equal to a square minus b square or we can say a square minus b square equals to 5.

02:52 Let's say this is equation 1.

02:56 Now, equate the imaginary part.

03:06 Imaginary parts we get 2 ab is equal to 12 therefore ab is equal to and hence b is equal to 6 upon a.

03:40 Substitute the value of b in equation 1 we get a square minus 6 by a square is equal to 5 and this will be equals to a square minus 36 upon a square is equal to 5 and we can write it as a to the power 4 minus 5 a square minus 36 is equal to 0 the factorization of this quality equation provides a square minus 9 into a square plus 4 is equal to 0.

04:42 From here we can see that a square plus 4 is equal to 0 is not possible since it will provide the imaginary value of a but it is given that a is real.

05:00 Therefore a square minus 9 is equal to 0 is possible and hence a square equals to 9 and a is equal to plus minus 3...

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