Math Problem Statement
6+i/(5−5i)^2 in standard form
Solution
To express in standard form, we need to follow these steps:
Step 1: Simplify the denominator
First, we calculate . To do this, use the formula for squaring a binomial: Let and . Now, apply the formula: Since , this simplifies to: Thus, .
Step 2: Multiply numerator and denominator by the conjugate of the denominator
Now we have: To simplify this, multiply both the numerator and denominator by the conjugate of the denominator, which is (since ).
Step 3: Simplify
First, distribute in the numerator: So the expression becomes:
Step 4: Write in standard form
The standard form of a complex number is , where and are real numbers. Therefore, the final expression is:
Thus, the standard form of is:
Would you like more details on any step or have questions?
Related questions:
- How do you find the conjugate of a complex number?
- What is the general formula for dividing complex numbers?
- How can you simplify powers of ?
- What are the properties of binomials involving complex numbers?
- How do you convert a complex number from polar to rectangular form?
Tip: When dividing complex numbers, multiplying by the conjugate of the denominator often simplifies the process.
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Math Problem Analysis
Mathematical Concepts
Complex Numbers
Algebra
Binomial Expansion
Formulas
Square of a binomial: (a - b)^2 = a^2 - 2ab + b^2
Multiplying by the conjugate: z / w = (z * conjugate(w)) / (w * conjugate(w))
Theorems
Properties of complex numbers: i^2 = -1
Division of complex numbers using conjugates
Suitable Grade Level
Grades 10-12