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Find the modulus of the given complex number 8i/(8-15i).

a) 17/√5
b) 8/√17
c) 15/√17
d) 8/√5

1 Answer

6 votes

Final answer:

The modulus of 8i/(8-15i) is calculated by multiplying the complex number by its conjugate and simplifying the expression. The final modulus is 8/√17, which corresponds to option (b).

Step-by-step explanation:

To find the modulus of a complex number, we can multiply the number by its conjugate and then take the square root of the result. The complex number in question is 8i/(8-15i). To eliminate the complex denominator, we multiply both the numerator and the denominator by the conjugate of the denominator, which is (8+15i). The result is:

(8i/(8-15i)) * ((8+15i)/(8+15i)) = (8i * (8+15i)) / ((8-15i) * (8+15i))

Expanding the numerator, we get: 64i + 120i², and since i² = -1, this simplifies to 64i - 120. The denominator, when expanded, becomes 8² - (15i)² which is 64 - (-225) and further simplifies to 289. So the complex number now is (-120+64i)/289.

To find the modulus, we take the square root of the sum of the squares of the real and imaginary parts:

Modulus = √((-120)² + (64)²) / √(289)

That gives us: √(14400 + 4096) / √(289) = √(18496) / 17 = 136 / 17 = 8.

Therefore, the modulus of 8i/(8-15i) is 8/√17, which matches answer option (b).

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