Problem 25 - Algebra 2 and Trigonometry

Question

In \(19-34,\) write each sum or difference in terms of \(i\) $$ \sqrt{4}+\sqrt{-4}+\sqrt{-36}-\sqrt{36} $$

Step-by-Step Solution

Answer

The expression simplifies to \(-4 + 8i\).

Step 1: Simplify Real Square Roots

Identify and simplify the real square roots in the expression. Consider \( \sqrt{4} \) and \( \sqrt{36} \):- \( \sqrt{4} = 2 \)- \( \sqrt{36} = 6 \)

Step 2: Simplify Imaginary Square Roots

Identify and simplify the imaginary square roots using the fact that \( \sqrt{-a} = \sqrt{a} \times i \):- \( \sqrt{-4} = \sqrt{4} \times i = 2i \)- \( \sqrt{-36} = \sqrt{36} \times i = 6i \)

Step 3: Substitute and Simplify the Expression

Substitute the simplified square roots back into the original expression and simplify:- Original problem: \( \sqrt{4} + \sqrt{-4} + \sqrt{-36} - \sqrt{36} \)- Substitute: \( 2 + 2i + 6i - 6 \)- Simplify: \( (2 - 6) + (2i + 6i) = -4 + 8i \)

Step 4: Write the Final Expression

The expression is now simplified in terms of \(i\):\(-4 + 8i\).

Key Concepts

Square Roots

Square roots are a fundamental concept in mathematics, representing a number that, when multiplied by itself, gives the original number. For instance, the square root of 4, denoted as \( \sqrt{4} \), is 2 because \( 2 \times 2 = 4 \). Regular square roots deal with positive numbers or zero, and this is because these numbers have real square roots.
When solving problems, it's often important to simplify square roots to make calculations easier. For example:

• \( \sqrt{4} = 2 \)
• \( \sqrt{16} = 4 \)
• \( \sqrt{36} = 6 \)

Square roots of negative numbers, however, lead us into the fascinating realm of complex numbers, which require a different approach.

Complex Numbers

Complex numbers expand our number system beyond the real numbers, allowing us to include the square roots of negative numbers. A complex number is expressed as \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit.
Consider complex numbers as a combination of real parts and imaginary parts:

• The real part, represented by \( a \), is a familiar real number.
• The imaginary part is \( bi \), where \( b \) is a real number and \( i \) represents the square root of \(-1\).

This combination allows complex numbers to describe concepts that ordinary real numbers cannot, effectively solving equations like \( x^2 + 1 = 0 \).The power of complex numbers lies in their ability to simplify the square roots of negative numbers, enabling calculations that involve such expressions. Using them can seem challenging at first, but with practice, they become an invaluable tool in mathematics.

Imaginary Unit

The imaginary unit \( i \) is the cornerstone of understanding complex numbers. Defined as the square root of \(-1\), \( i \) is not a typical number you find on the number line. Instead, it is a foundational element that extends the concept of square roots to negative values.
Key points about the imaginary unit include:

• \( i^2 = -1 \)
• To find the square root of a negative number, such as \( \sqrt{-a} \), you can express it as \( \sqrt{a} \times i \).

For example, when tackling the square root of \(-4\), you rewrite it using the imaginary unit:
\( \sqrt{-4} = \sqrt{4} \times i = 2i \). This principle allows for the simplification of expressions involving the square roots of negative numbers, as seen in complex number calculations.Understanding \( i \) not only demystifies the concept of negative square roots but also unlocks an entire dimension of mathematical applications in fields ranging from engineering to physics.