Question 3 of 28beginner🔧 ApplyNumerical1 mark

Express the following complex number in the form a+iba + ib: i−39i^{-39}

Correct Answer

i

Exercise: EXERCISE 4.1 | Q: 3 | (Page 7)
For More Understanding

Explanation

The problem requires simplifying a negative power of ii. Following the method in Example 6(ii), we first convert the negative exponent to a positive one in the denominator. We then simplify the power of ii using the property i2=−1i^2 = -1 or the cyclic nature of powers of ii (i4k+3=−ii^{4k+3} = -i). Finally, we rationalize the denominator to express the result in the standard form a+iba + ib.

Solution Steps

  1. Step 1: Convert negative exponent to fraction: i−39=1i39i^{-39} = \frac{1}{i^{39}}

  2. Step 2: Simplify the power of ii: We write i39=i38⋅i=(i2)19⋅ii^{39} = i^{38} \cdot i = (i^2)^{19} \cdot i. Substituting i2=−1i^2 = -1, we get (−1)19⋅i=−1⋅i=−i(-1)^{19} \cdot i = -1 \cdot i = -i.

  3. Step 3: Substitute back into the fraction: i−39=1−ii^{-39} = \frac{1}{-i}

  4. Step 4: Rationalize the denominator: Multiply the numerator and denominator by ii.

    1−i×ii=i−i2\frac{1}{-i} \times \frac{i}{i} = \frac{i}{-i^2}

  5. Step 5: Final simplification: Since i2=−1i^2 = -1, we have i−(−1)=i1=i\frac{i}{-(-1)} = \frac{i}{1} = i.

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