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Chapter 6: Problem 8

Write each expression in the form \(a+b i,\) where \(a\) and \(b\) are real numbers. $$ (5+6 i)(2+7 i) $$

Short Answer

Expert verified
The short answer is \(-32 + 47i\).

Step by step solution

01

Write down the given expression

We are given the complex numbers expression: \[ (5+6i)(2+7i) \]
02

Apply the distributive property

Now, multiply each part of the first complex number by each part of the second one, as we would do it for binomials: \[ (5+6i)(2+7i) = 5 \cdot 2 + 5 \cdot 7i + 6i \cdot 2 + 6i \cdot 7i \]
03

Perform the multiplications

Next, perform each multiplication: \[ = 10 + 35i + 12i + 42i^2 \]
04

Recall the definition of "i"

Recall that \(i\) is the imaginary unit and is defined as \(i^2 = -1\). Substitute this definition into the expression: \[ = 10 + 35i + 12i - 42 \]
05

Simplify the real and imaginary parts

Lastly, combine the real and imaginary parts into one expression in the form a + bi: \[ = (10 - 42) + (35i + 12i) = -32 + 47i \] The final expression in the form a + bi is \(-32 + 47i\).

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Most popular questions from this chapter

Suppose \(w\) and \(z\) are complex numbers, with \(z \neq 0\). Show that \(\overline{\left(\frac{w}{z}\right)}=\frac{\bar{w}}{\bar{z}}\).

Suppose \(z\) is a complex number. Show that \(\frac{z-\bar{z}}{2 i}\) equals the imaginary part of \(z\).

Write $$ \left(\cos \frac{\pi}{7}+i \sin \frac{\pi}{7}\right)\left(\cos \frac{\pi}{9}+i \sin \frac{\pi}{9}\right) $$ in polar form.

Show that addition and multiplication of complex numbers satisfy the distributive property, meaning that $$ u(w+z)=u w+u z $$ for all complex numbers \(u, w,\) and \(z\).

Find two complex numbers \(z\) that satisfy the equation \(2 z^{2}+4 z+5=0\).
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