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Schach [20]
3 years ago
8

Can some one please explain this? Thank you all!

Mathematics
1 answer:
IRINA_888 [86]3 years ago
8 0
Basically bill wrote or chose a number. That number that he chose needed to have 2 things.

1. It needed to have been a number in between 27 and 45

2. It needed to have been an odd number in between 27 and 45.

There are only 2 possible choices that are odd, so eliminate the other 2 that are even. Next look at the 2 numbers that are odd.

The digits for each number must add up to 8.

35 - 3 + 5 = 8
43 - 4 + 3 = 7.

Which number adds up to 8. The digits in 35 or 43? It is 35.

Thus the final solution is 35. The number that Bill wrote.
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Given <em>z</em> = 3 + <em>i</em>, right away we can find

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<em>z</em> ² = (3 + <em>i </em>)² = 3² + 6<em>i</em> + <em>i</em> ² = 9 + 6<em>i</em> - 1 = 8 + 6<em>i</em>

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Any complex number has 2 square roots. Using the polar form from part (d), we have

√<em>z</em> = √(√10) exp(<em>i</em> arctan(1/3) / 2)

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Then in standard rectangular form, we have

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\sqrt z = \sqrt[4]{10} \left(\cos\left(\dfrac12 \arctan\left(\dfrac13\right) + \pi\right) + i \sin\left(\dfrac12 \arctan\left(\dfrac13\right) + \pi\right)\right)

We can simplify this further. We know that <em>z</em> lies in the first quadrant, so

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\cos\left(\dfrac12 \arctan\left(\dfrac13\right)+\pi\right) = -\sqrt{\dfrac{1+\cos\left(\arctan\left(\dfrac13\right)\right)}2}

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