Math Problem Statement

13i/10i(5-3i)

Solution

Let's simplify the expression:

13i10i(53i)\frac{13i}{10i(5 - 3i)}

Step 1: Simplify the denominator

Multiply the denominator:

10i(53i)=10i510i3i=50i30i210i(5 - 3i) = 10i \cdot 5 - 10i \cdot 3i = 50i - 30i^2

Recall that i2=1i^2 = -1, so:

50i30(1)=50i+30=30+50i50i - 30(-1) = 50i + 30 = 30 + 50i

Thus, the expression becomes:

13i30+50i\frac{13i}{30 + 50i}

Step 2: Rationalize the denominator

To remove the imaginary part from the denominator, multiply both the numerator and the denominator by the conjugate of the denominator, 3050i30 - 50i:

13i30+50i×3050i3050i=13i(3050i)(30+50i)(3050i)\frac{13i}{30 + 50i} \times \frac{30 - 50i}{30 - 50i} = \frac{13i(30 - 50i)}{(30 + 50i)(30 - 50i)}

Step 3: Multiply the numerator and simplify

First, let's expand the numerator:

13i(3050i)=13i3013i50i=390i650i213i(30 - 50i) = 13i \cdot 30 - 13i \cdot 50i = 390i - 650i^2

Since i2=1i^2 = -1:

390i650(1)=390i+650=650+390i390i - 650(-1) = 390i + 650 = 650 + 390i

Step 4: Multiply the denominator

Now expand the denominator:

(30+50i)(3050i)=302(50i)2=9002500i2(30 + 50i)(30 - 50i) = 30^2 - (50i)^2 = 900 - 2500i^2

Since i2=1i^2 = -1:

9002500(1)=900+2500=3400900 - 2500(-1) = 900 + 2500 = 3400

Step 5: Final answer

Now, the expression becomes:

650+390i3400\frac{650 + 390i}{3400}

We can simplify this by dividing both the real and imaginary parts by 3400:

6503400+390i3400=65340+39i340=1368+39i340\frac{650}{3400} + \frac{390i}{3400} = \frac{65}{340} + \frac{39i}{340} = \frac{13}{68} + \frac{39i}{340}

Thus, the simplified form of the expression is:

1368+39i340\frac{13}{68} + \frac{39i}{340}

Do you want more details or have any questions?

Here are 5 related questions to practice:

  1. Simplify 5i2i(4i)\frac{5i}{2i(4 - i)}.
  2. Rationalize and simplify 3+2i1i\frac{3 + 2i}{1 - i}.
  3. Find the conjugate of 4+i32i\frac{4 + i}{3 - 2i}.
  4. Simplify 2+3i1+2i\frac{2 + 3i}{1 + 2i}.
  5. Multiply (2+i)(34i)(2 + i)(3 - 4i) and simplify.

Tip: Always remember that i2=1i^2 = -1 when simplifying complex numbers.

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Rationalization
Multiplication of Complex Numbers

Formulas

i^2 = -1
Rationalizing complex numbers by multiplying by the conjugate: (a + bi)(a - bi) = a^2 - b^2i^2

Theorems

Rationalization of Complex Numbers

Suitable Grade Level

Grades 10-12