Math Problem Statement

6+i/(5−5i)^2 in standard form

Solution

To express 6+i(55i)2\frac{6 + i}{(5 - 5i)^2} in standard form, we need to follow these steps:

Step 1: Simplify the denominator

First, we calculate (55i)2(5 - 5i)^2. To do this, use the formula for squaring a binomial: (ab)2=a22ab+b2(a - b)^2 = a^2 - 2ab + b^2 Let a=5a = 5 and b=5ib = 5i. Now, apply the formula: (55i)2=522(5)(5i)+(5i)2(5 - 5i)^2 = 5^2 - 2(5)(5i) + (5i)^2 =2550i+25i2= 25 - 50i + 25i^2 Since i2=1i^2 = -1, this simplifies to: 2550i+25(1)=2550i25=50i25 - 50i + 25(-1) = 25 - 50i - 25 = -50i Thus, (55i)2=50i(5 - 5i)^2 = -50i.

Step 2: Multiply numerator and denominator by the conjugate of the denominator

Now we have: 6+i50i\frac{6 + i}{-50i} To simplify this, multiply both the numerator and denominator by the conjugate of the denominator, which is ii (since ii=1i \cdot i = -1).

(6+i)i50ii=(6+i)i50(1)\frac{(6 + i) \cdot i}{-50i \cdot i} = \frac{(6 + i)i}{-50(-1)}

Step 3: Simplify

First, distribute ii in the numerator: (6+i)i=6i+i2=6i1(6 + i)i = 6i + i^2 = 6i - 1 So the expression becomes: 6i150=150+6i50\frac{6i - 1}{50} = \frac{-1}{50} + \frac{6i}{50}

Step 4: Write in standard form

The standard form of a complex number is a+bia + bi, where aa and bb are real numbers. Therefore, the final expression is: 150+6i50=150+3i25\frac{-1}{50} + \frac{6i}{50} = -\frac{1}{50} + \frac{3i}{25}

Thus, the standard form of 6+i(55i)2\frac{6 + i}{(5 - 5i)^2} is: 150+3i25-\frac{1}{50} + \frac{3i}{25}

Would you like more details on any step or have questions?

Related questions:

  1. How do you find the conjugate of a complex number?
  2. What is the general formula for dividing complex numbers?
  3. How can you simplify powers of ii?
  4. What are the properties of binomials involving complex numbers?
  5. 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