Question

Perform the indicated operation and write the result in standard form. $$(-2+\sqrt{-8})+(5-\sqrt{-50})$$

Short answer

The answer is \(3 - 3i\sqrt{2}\)

Step-by-step solution

Simplify Square Roots

Begin by simplifying each square root. Remember that \(\sqrt{-1}\) is represented by \(i\). So, \(\sqrt{-8}\) equals \(2i\sqrt{2}\) and \(\sqrt{-50}\) equals \(5i\sqrt{2}\). Now, the expression can be rewritten as follows: \(-2 + 2i\sqrt{2} + 5 - 5i\sqrt{2}\)

Combine Like Terms

Next, combine like terms - real with real and imaginary with imaginary, yielding \(-2 + 5 + 2i\sqrt{2} - 5i\sqrt{2}\).

Fluid Final Form

Finally, you can simplify this to get it in standard form: \(3 - 3i\sqrt{2}\)

Key concepts

Simplifying Square Roots

When dealing with square roots in algebra, particularly those involving negative numbers, it’s essential to understand how to simplify them correctly. The square root of a negative number introduces the concept of imaginary numbers. In the context of simplifying square roots, recall that \( \(\sqrt{-1}\) \) is represented as \(i\), which is the fundamental imaginary unit.

For example, to simplify \(\sqrt{-8}\), one would first separate the square root of the negative number into two parts: \(\sqrt{8}\) and \(\sqrt{-1}\). The square root of 8 can be further simplified to \(2\sqrt{2}\) since 8 is 4 times 2 and 4 is a perfect square. Coupling this with the imaginary unit \(i\) yields \(2i\sqrt{2}\), which is the simplified form of \(\sqrt{-8}\). Similarly, \(\sqrt{-50}\) simplifies to \(5i\sqrt{2}\) after realizing that 50 is 25 times 2, and 25 is a perfect square. It’s crucial to identify perfect squares and simplify them to reduce the square roots to their most basic form.

The process of simplifying square roots can be summarized in a few steps:

• Identify and extract any perfect square factors from under the square root.
• Separate the negative component and represent it with an \(i\).
• Combine these simplified components to express the square root in its simplest form.

Imaginary Numbers

Imaginary numbers are a crucial concept in algebra, especially when working with complex numbers. The term 'imaginary' may sound elusive, but these numbers are just as 'real' as real numbers in mathematics. An imaginary number is defined as any real number multiplied by \(i\), which is the square root of -1.

In our context, when simplifying square roots of negative numbers, we encounter imaginary numbers. For instance, \(\sqrt{-1}\) becomes \(i\), and \(\sqrt{-8}\) can be written as \(2i\sqrt{2}\), as discussed in the previous section. It's important to note that the standard form for a complex number is \(a + bi\), where \(a\) and \(b\) are real numbers.

The presence of \(i\) distinguishes a complex number from a pure real number. When performing operations with imaginary numbers, remember to treat \(i\) just like any variable, until it’s time to simplify. This approach makes operations with complex numbers, involving simplification of square roots of negative numbers, entirely manageable.

• Recognize that every square root of a negative number can be represented as a product involving \(i\).
• Use this representation to perform operations as if \(i\) were a variable.
• Recombine terms to form a complex number in standard form.

Combining Like Terms

The next step after simplifying individual terms in an algebraic expression is to combine like terms. This process is akin to 'tidying up' the equation, where you bring together terms that are alike—terms that share the same variables and powers.

When combining like terms that contain imaginary numbers, remember to group real numbers (those without \(i\)) and imaginary numbers (those with \(i\)) separately. Consider the expression \( -2 + 2i\sqrt{2} + 5 - 5i\sqrt{2} \). Here, \( -2 \) and \( +5 \) are like terms because they are both real numbers. Similarly, \( 2i\sqrt{2} \) and \( -5i\sqrt{2} \) are like terms because they are both imaginary numbers with the same \(\sqrt{2}\) element.

By adding together the real numbers and the imaginary numbers separately, the simplified expression becomes:

• Combine real numbers: \( -2 + 5 = 3 \).
• Combine imaginary numbers: \( 2i\sqrt{2} - 5i\sqrt{2} = -3i\sqrt{2} \).
• Combine these results to form the complex number in standard form: \( 3 - 3i\sqrt{2} \).

This organized approach ensures that you simplify the expression correctly and avoid mixing different types of terms, which could lead to errors.