Question

Simplify the complex numbers and arrange them in increasing order of their absolute value (modulus).

1) (sqrt3-sqct the correct answer. convert 3cis 180° to rectangular form.rt3i)⁴
2) (-1 sqrt3i)¹²
3) (sqrt 3-i)⁶
4) (sqrt2-sqrt2i)⁸
5) (sqrt2-i)⁶

Answer 1

To answer the student's question, we simplify each given complex number expression, calculate its absolute value, and then arrange them in increasing order of their modulus. This process involves using algebraic identities and properties of complex numbers.

Step-by-step explanation:

The question involves simplifying complex numbers and arranging them in increasing order of their absolute value (modulus). The absolute value of a complex number is calculated using the formula |A| = √(a² + b²), where a and b are the real and imaginary parts of the complex number A, respectively. Using this formula and relevant algebraic identities, each given complex expression can be simplified to find its absolute value.

Now, to address the given expressions:

1. (√3-√3i)⁴ simplifies to a real number and its absolute value can be calculated.
2. (-1+√3i)¹² simplifies, taking into account that the powers of i repeat every four terms.
3. (√3-i)⁶ can be simplified using binomial expansion and the properties of i.
4. (√2-√2i)⁸ simplifies, also considering the properties of i.
5. (√2-i)⁶ can be handled similarly with binomial expansion.

Once simplified, the absolute values can be compared and the complex numbers arranged accordingly.