Question Video: Simplifying a Complex Number in Exponential Form | Nagwa Question Video: Simplifying a Complex Number in Exponential Form | Nagwa

# Question Video: Simplifying a Complex Number in Exponential Form Mathematics • Third Year of Secondary School

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Express the complex number π§ = π^(β4 β (23π/12 π) in the form of π β π^(ππ).

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### Video Transcript

Express the complex number π§ is equal to π to the power of negative four minus 23π over 12 π in the form of π multiplied by π to the power of ππ.

In this question, weβre asked to express the complex number in exponential form such that π§ is equal to π multiplied by π to the power of ππ, where π is the magnitude, or modulus, of the complex number and π is its argument.

We will begin by defining π in terms of its principal value, that is, the value of π that is greater than negative π and less than or equal to π. Letβs go back to our complex number and see how we can write it in this form. We recall from our laws of exponents or indices that π₯ to the power of π multiplied by π₯ to the power of π is equal to π₯ to the power of π plus π. This means that we can rewrite the complex number as π to the power of negative four multiplied by π to the power of negative 23π over 12 π.

We now have our value for π. Itβs π to the power of negative four. π is negative 23π over 12. However, we want our principal value to be greater than negative π and less than or equal to π. And clearly, our value is outside of this range. We recall that we can find the principal value by adding or subtracting multiples of two π to the value of π. In this case, we will add two π to negative 23π over 12. We can write two π as 24π over 12, giving us negative 23π over 12 plus 24π over 12. This is equal to π over 12. And we can therefore express our complex number as π to the power of negative four multiplied by π to the power of π over 12 π. This is the exponential form of the complex number π to the power of negative four minus 23π over 12 π.

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