Test 8 (Lessons 15-16): Complex Numbers Solutions

What is the definition of the imaginary number, i ?

The imaginary number i is defined as $i^{2} = - 1$ .

Evaluate.

$i^{37}$

$i^{36} \cdot i {(i^{4})}^{9} \cdot i 1 \cdot i$

$i^{79}$

$i^{76} \cdot i^{3} {(i^{4})}^{19} \cdot i^{3} 1 \cdot (- i)$

– i

Simplify in the form $a + b i .$
Name all of the sets to which the problems belong: real complex, pure imaginary complex, or complex.

$(5 i - 36) + (25 - 8 i)$

$- 11 - 3 i$

–11 + 3i
complex

$4 (3 + 2 i^{2})$

$12 + 8 i^{2} 12 + 8 (- 1)$

4
real complex

$(6 i - 3) (i^{8} + i)$

$6 i^{9} + 6 i^{2} - 3 i^{8} - 3 i (6 \cdot i^{8} \cdot i) + 6 (- 1) - (3 \cdot {(i^{4})}^{2}) - 3 i (6 \cdot {(i^{4})}^{2} \cdot i) - 6 - (3 \cdot 1) - 3 i (6 \cdot 1 \cdot i) - 6 - 3 - 3 i 6 i - 9 - 3 i$

–9 + 3i
complex

$12 i - (3 i^{3} + 5 i^{3})$

$12 i - (8 i^{3}) 12 i - 8 (- i)$

20i
pure imaginary, complex

Simplify using imaginary numbers.

$\frac{3 i}{\sqrt{- 50}}$

$\sqrt{- 50} = i \sqrt{2 \cdot 5^{2}} = 5 i \sqrt{2} \frac{3 i}{5 i \sqrt{2}} (\frac{i \sqrt{2}}{i \sqrt{2}}) \frac{3 i^{2} \sqrt{2}}{5 i^{2} (2)} \frac{3 (- 1) \sqrt{2}}{5 (- 1) (2)}$

$\frac{3 \sqrt{2}}{10}$

$\frac{- 2 i}{4 - 5 i}$

$\frac{- 2 i}{4 - 5 i} (\frac{4 + 5 i}{4 + 5 i}) \frac{- 8 i - 10 i^{2}}{16 - 25 i^{2}} \frac{- 8 i - 10 (- 1)}{16 - 25 (- 1)}$

$\frac{10 - 8 i}{41}$

Determine the value of M that forms a polynomial identity.
${(3 x + i)}^{2} + (M - 6 i x) = (3 x - 2 i) (3 x + 2 i)$

$9 x^{2} + 3 i x + 3 i x + i^{2} + M - 6 i x 9 x^{2} + 6 i x + (- 1) + M - 6 i x 9 x^{2} - 1 + M - 1 + M = 4$

im $9 x^{2} + 6 i x - 6 i x - 4 i^{2} 9 x^{2} - 4 (- 1) 9 x^{2} + 4$

M = 5