Express the following in the form of a -ib-2-1-2-1-3 i3

( 2 1( 2 1/3i ))^3 = { 2 1 (6i 1/3i) }^3 = { 6i 6i+1/3i}^3 = {1 12i/3i}^3 = (1 2i)^3/9i^3 = 1 8i^3 3(1)(2i)(1 2i)/ 9i (∵ i^3 = i) = 1 8i^3 6i +12i^2/ ...

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Byju's Answer

Standard XII

Mathematics

Implicit Differentiation

Express the f...

Question

Express the following in the form of a +ib:
$(- 2 - 1 (- 2 - \frac{1}{3 i}))^{3}$

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Solution

$(- 2 - 1 (- 2 - \frac{1}{3 i}))^{3} = {- 2 - 1 (\frac{6 i - 1}{3 i})}^{3} = {\frac{- 6 i - 6 i + 1}{3 i}}^{3} = {\frac{1 - 12 i}{3 i}}^{3} = \frac{(1 - 2 i)^{3}}{9 i^{3}} = \frac{1 - 8 i^{3} - 3 (1) (2 i) (1 - 2 i)}{- 9 i} (∵ i^{3} = - i) = \frac{1 - 8 i^{3} - 6 i + 12 i^{2}}{- 9 i} = \frac{+ 8 i - 6 i + 1 - 12}{- 9 i} \frac{2 i - 11}{- 9 i} = \frac{(2 i - 11) (9 i)}{(- 9 i) (9 i)} = \frac{18 i^{2} - 19 i}{- 81 i^{2}} = \frac{- 18 - 99 i}{81} = - \frac{18}{81} \div \frac{99}{81} i = - \frac{2}{9} - \frac{11}{9} i$

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Express the following in the form of a +ib: (−2−1(−2−13i))3

(−2−1(−2−13i))3={−2−1(6i−13i)}3={−6i−6i+13i}3={1−12i3i}3=(1−2i)39i3=1−8i3−3(1)(2i)(1−2i)−9i (∵i3=−i)=1−8i3−6i+12i2−9i=+8i−6i+1−12−9i 2i−11−9i=(2i−11)(9i)(−9i)(9i)=18i2−19i−81i2=−18−99i81=−1881÷9981i=−29−119i

Express the following in the form of a +ib:
$(- 2 - 1 (- 2 - \frac{1}{3 i}))^{3}$