Hi,

I'd be very appreciative if someone could explain the steps and methodology to 'evaluate' the following:

1) Evaluate

i) $\displaystyle (2 + 3i)(2 - 3i)$

ii) $\displaystyle 7 - 3i / 2 + 4i$

iii) $\displaystyle i^5(1 + i^5)$

Many regards,

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- Aug 25th 2009, 06:47 PMrel85i evaluations
Hi,

I'd be very appreciative if someone could explain the steps and methodology to 'evaluate' the following:

1) Evaluate

i) $\displaystyle (2 + 3i)(2 - 3i)$

ii) $\displaystyle 7 - 3i / 2 + 4i$

iii) $\displaystyle i^5(1 + i^5)$

Many regards, - Aug 25th 2009, 07:17 PMartvandalay11
i) $\displaystyle (2 + 3i)(2 - 3i)=2(2)+2(-3i)+3i(2)+3i(-3i)$ we just foiled

$\displaystyle =4-6i+6i-9i^2=4-9(-1)=4+9=13$ and since $\displaystyle i=\sqrt{-1}, i^2=-1$

Should there be parenthesis for the fraction in ii)? otherwise just combine the i terms

iii) $\displaystyle i^5(1 + i^5)$

$\displaystyle i^5=i^4*i=i^2*i^2*i=(-1)(-1)i=i$ so we now have

$\displaystyle i(1+i)=i+i^2=i-1=-1+i$

In general, to evaluate $\displaystyle i^n$ divide n by 4, and take the remainder as the power, for instance $\displaystyle i^{27}$ $\displaystyle \frac{27}{4}=6$ remainder 3, so $\displaystyle i^{27}=i^3=-i$

$\displaystyle i^0=1$

$\displaystyle i^1=i$

$\displaystyle i^2=-1$

$\displaystyle i^3=-i$

$\displaystyle i^4=1$ this is why we divide by 4, the cycle repeats as you see here - Aug 25th 2009, 07:40 PMrel85
Am I doing something wrong for the first one?

I get $\displaystyle 4 - 9i^2 + 6^i - 6^i$ from expanding the brackets which becomes $\displaystyle 4 - 9i^2 = 4 - 9(-1) == 13$

Does it stop there, where is the i^2 = -1 coming from at this point?

No parenthesises for the second, just a fraction... Group like terms how?

Thanks! - Aug 25th 2009, 07:46 PMKrahl
$\displaystyle

7 - 3i / 2 + 4i

$

$\displaystyle 7-\frac{3}{2}i+4i$

$\displaystyle 7+(4-\frac{3}{2})i$

group the i terms as you would x terms - Aug 25th 2009, 07:46 PMartvandalay11
$\displaystyle (\sqrt{x})^2=x$

$\displaystyle i=\sqrt{-1}$ so square both sides

$\displaystyle i^2=(\sqrt{-1})^2=-1$

$\displaystyle 7-\frac{3i}{2}+4i=7-\frac{3i}{2}+\frac{8i}{2}$

$\displaystyle =7+\frac{5i}{2}$