Express the following in the form $a+i b$ $$\begin{array}{ll} \text { (i) } \frac{5+\sqrt{2} i}{1-\sqrt{2} i} & \text { (ii) } i^{-35} \end{array}$$

Step 1: Simplify the first expression (i)

To express a complex number in the form a+ib, when it is given as a fraction, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of $$1-\sqrt{2}i$$ is $$1+\sqrt{2}i$$ .

Step 2: Expand the numerator and denominator

We expand both the numerator and the denominator. In the denominator, we use the identity $$(x-y)(x+y) = x^2 - y^2$$ . Remember that $$i^2 = -1$$ .

Step 3: Simplify the expression

Substitute $$i^2 = -1$$ into the expression. Combine the real and imaginary terms in the numerator and simplify the denominator.

Step 4: Final form for (i)

Divide each term in the numerator by the denominator to get the expression in the form a+ib. Here, $$a=1$$ and $$b=2\sqrt{2}$$ .

Step 5: Simplify the second expression (ii)

For the second expression, we first rewrite $$i^{-35}$$ as $$\frac{1}{i^{35}}$$ .

Step 6: Simplify the power of i

We know that the powers of $$i$$ repeat in a cycle of 4: $$i^1=i$$ , $$i^2=-1$$ , $$i^3=-i$$ , $$i^4=1$$ . To simplify $$i^{35}$$ , we divide the exponent by 4 and use the remainder. $$35 = 4 \times 8 + 3$$ , so $$i^{35} = i^3 = -i$$ .

Step 7: Substitute and rationalize

Substitute $$i^{35} = -i$$ back into the expression. To remove $$i$$ from the denominator, multiply the numerator and denominator by $$i$$ . Remember $$i^2 = -1$$ .

Step 8: Final form for (ii)

The expression $$i$$ can be written in the form a+ib as $$0+1i$$ . Here, $$a=0$$ and $$b=1$$ .

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