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Chapter 2: Problem 55

Write the quotient in standard form. $$\frac{5+i}{5-i}$$

Short Answer

Expert verified
The solution for the quotient is \(12/13 + (5i/13)\)

Step by step solution

01

Identify the Conjugate

The conjugate of a complex number \(a + bi\) is \(a - bi\). In this case, the complex number in the denominator is 5 - i, so the conjugate is 5 + i.
02

Multiply by the Conjugate

Multiply the numerator and denominator of the fraction \((5 + i) / (5 - i)\) by the conjugate of the denominator. This will look like \((5 + i) / (5 - i) \times (5 + i) / (5 + i)\). This yields \((5 + i) ^2 / (5 - i)(5 + i)\).
03

Simplify the expression

To simplify the expression, carry out the multiplication in the numerator and denominator which results in: \The numerator simplification: \( (5 + i) ^2 = 25 + 10i + i^2 \). \The denominator simplification: \( (5 - i)(5 + i) = 25 - i^2 \). \Remember that \(i^2 = -1\). So the full simplified fraction is \( (25 + 10i - 1) / (25 + 1) = (24 + 10i) / 26 \).
04

Write in Standard Form

To get the result in standard form, divide the real part and the imaginary part by the real number in the denominator thus getting \(24/26 + (10i/26)\), which simplifies to \(12/13 + (5i/13)\) . It's now in standard form.

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Most popular questions from this chapter

Prove that the complex conjugate of the product of two complex numbers \(a_{1}+b_{1} i\) and \(a_{2}+b_{2} i\) is the product of their complex conjugates.

(a) Find the interval(s) for \(b\) such that the equation has at least one real solution and (b) write a conjecture about the interval(s) based on the values of the coefficients. $$3 x^{2}+b x+10=0$$

Write the polynomial as the product of linear factors and list all the zeros of the function. $$g(x)=2 x^{3}-x^{2}+8 x+21$$

Find all the zeros of the function. When there is an extended list of possible rational zeros, use a graphing utility to graph the function in order to disregard any of the possible rational zeros that are obviously not zeros of the function. $$f(x)=16 x^{3}-20 x^{2}-4 x+15$$

Find all real zeros of the function. $$f(x)=4 x^{3}-3 x-1$$
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