Simplify Complex Numbers: (7 + √-9)(5 + √-36)

by Alex Johnson 46 views

When you encounter a math problem that asks you to simplify complex numbers and write them in the standard form of a + bi, it might seem a little intimidating at first glance. However, with a clear understanding of the properties of imaginary numbers and a systematic approach, these problems become quite manageable. This article will walk you through simplifying the expression (7 + √-9)(5 + √-36), breaking down each step so you can confidently tackle similar problems in the future.

Understanding Imaginary Numbers

Before we dive into the simplification, let's quickly recap what imaginary numbers are. The cornerstone of imaginary numbers is the imaginary unit, denoted by i, which is defined as the square root of -1 (i.e., i = √-1). Consequently, i² = -1. This fundamental concept allows us to work with the square roots of negative numbers. For any positive real number 'a', the square root of -a can be expressed as √-a = √a * √-1 = √a * i. Keeping this in mind is crucial for simplifying terms like √-9 and √-36 within our problem.

Simplifying the Square Roots

Our first step in simplifying (7 + √-9)(5 + √-36) is to rewrite the square roots of negative numbers in terms of i. Let's tackle each one separately:

  • √-9: We can break this down as √(-1 * 9). Using the property of square roots, this becomes √-1 * √9. We know that √-1 = i and √9 = 3. Therefore, √-9 = 3i.
  • √-36: Similarly, we can express this as √(-1 * 36). This leads to √-1 * √36. Since √-1 = i and √36 = 6, we have √-36 = 6i.

Now, we can substitute these simplified forms back into our original expression:

(7 + 3i)(5 + 6i)

This expression is now in a more familiar form, ready for multiplication.

Multiplying Complex Numbers

Multiplying two complex numbers in the form (a + bi)(c + di) is very similar to multiplying two binomials, often remembered by the acronym FOIL (First, Outer, Inner, Last). We distribute each term in the first complex number to each term in the second complex number:

  • First: Multiply the first terms: a * c
  • Outer: Multiply the outer terms: a * di
  • Inner: Multiply the inner terms: bi * c
  • Last: Multiply the last terms: bi * di

Combining these, we get: ac + adi + bci + bdi².

Since i² = -1, the expression becomes: ac + adi + bci - bd.

Finally, we group the real parts (terms without 'i') and the imaginary parts (terms with 'i') to express the result in the form a + bi:

(ac - bd) + (ad + bc)i

Now, let's apply this to our specific problem: (7 + 3i)(5 + 6i).

Applying the FOIL Method

Let's use the FOIL method to multiply (7 + 3i) by (5 + 6i):

  1. First: Multiply the first terms: 7 * 5 = 35
  2. Outer: Multiply the outer terms: 7 * 6i = 42i
  3. Inner: Multiply the inner terms: 3i * 5 = 15i
  4. Last: Multiply the last terms: 3i * 6i = 18i²

Combining these results, we get: 35 + 42i + 15i + 18i².

Simplifying the Result

Our next step is to simplify the expression 35 + 42i + 15i + 18i² by combining like terms and substituting i² = -1.

First, let's combine the imaginary terms (the terms with 'i'):

42i + 15i = 57i

So, the expression becomes: 35 + 57i + 18i².

Now, we substitute i² = -1:

18i² = 18 * (-1) = -18

Substitute this back into our expression:

35 + 57i - 18

Finally, we combine the real terms (the constant numbers):

35 - 18 = 17

So, the simplified expression is 17 + 57i.

This result is in the standard a + bi form, where a = 17 and b = 57. Therefore, the correct answer among the options provided is A. 17 + 57i.

Conclusion

Simplifying expressions involving complex numbers, such as (7 + √-9)(5 + √-36), requires a solid understanding of imaginary numbers and their properties. By breaking down the problem into smaller, manageable steps – simplifying the square roots, applying the FOIL method for multiplication, and then combining like terms – we can arrive at the correct answer in the standard a + bi form. Remember that i = √-1 and i² = -1 are your most important tools in these calculations. Practice with various examples will further solidify your understanding and build your confidence in solving complex number problems.

For further exploration and resources on complex numbers, you can visit Khan Academy's section on Complex Numbers.