Math Problem Statement

Jika z = 2 - √(-9), maka |z^3| = ....

Solution

The expression given in the image z=29z = 2 - \sqrt{-9} is a complex number. First, we'll simplify this expression and then find the modulus of its cube.

  1. Simplifying 9\sqrt{-9}: 9=3i\sqrt{-9} = 3i where ii is the imaginary unit with i2=1i^2 = -1.

  2. Express zz: z=23iz = 2 - 3i

  3. Calculating z3|z|^3: The modulus z|z| of a complex number z=a+biz = a + bi is calculated as: z=a2+b2|z| = \sqrt{a^2 + b^2} Applying this to our zz: z=22+(3)2=4+9=13|z| = \sqrt{2^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13} Then, the cube of the modulus z3|z|^3 is: z3=(13)3=133/2|z|^3 = (\sqrt{13})^3 = 13^{3/2}

Thus, z3=133/2|z^3| = 13^{3/2}.

Would you like to know more about complex numbers or any other related topics? Here are some additional questions you could ask:

  1. How do I represent complex numbers in the complex plane?
  2. What are the properties of modulus and argument in complex numbers?
  3. How can I multiply and divide complex numbers?
  4. What is the significance of Euler's formula in relation to complex numbers?
  5. Could you show how to solve quadratic equations that have complex solutions?

Tip: Remember, the modulus of a complex number gives you the distance of the point from the origin in the complex plane, which can be visualized as the hypotenuse of a right triangle formed by the real and imaginary parts.

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Modulus
Powers of Complex Numbers

Formulas

Modulus of complex number |z| = √(a^2 + b^2), where z = a + bi
Cube of the modulus |z^3| = |z|^3

Theorems

Modulus theorem for complex numbers

Suitable Grade Level

Grades 10-12