Problem 32 - College Algebra and Calculus: An Applied Approach

Question

Perform the indicated operation and write the result in standard form. $$(\sqrt{-75})^{3}$$

Step-by-Step Solution

Answer

The result of the operation \((\sqrt{-75})^{3}\) in standard form is \(-3750\sqrt{3}i\).

Step 1: Handle The Negative Under the Root

First step is to handle the square root of a negative number. The square root of -1 is represented as 'i' in complex numbers. So, \(\sqrt{-75}\) can be written as \(\sqrt{75} * \sqrt{-1}\), which is equivalent to \(5\sqrt{3}i\).

Step 2: Apply the Exponent

Next step is to apply the exponent 3 to \(5\sqrt{3}i\). When applying the exponent to a product, it can be individually applied to each term. So, \((5\sqrt{3}i)^3\) turns into \(125\sqrt{27}i^3\).

Step 3: Simplify the complex number

The expression \(i^3\) simplifies to -i. Hence, \(125\sqrt{27}i^3\) simplifies down to \(-125\sqrt{27}i\). Lastly, simplify \(125\sqrt{27}\) further to obtain \(-3750\sqrt{3}i\).

Key Concepts

Standard Form of a Complex Number

Understanding the standard form of a complex number is fundamental when working with complex algebra. The standard form of a complex number is expressed as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part, with \(i\) being the imaginary unit. This form allows us to easily perform arithmetic operations, including addition, subtraction, multiplication, and division.

When dealing with square roots of negative numbers or higher exponents, as in our exercise \((\sqrt{-75})^3\), restructuring the expression to follow this standard form is crucial. After simplifying the expression and resolving the imaginary parts, your final answer should still align with the \(a + bi\) format. This not only ensures correctness but also facilitates better understanding and further calculations with complex numbers.

Negative Square Root

The square root of a negative number is a common challenge when first learning about complex numbers. By definition, you cannot have a real number which, when squared, equals a negative number. The concept of an 'imaginary unit' designated as \(i\) comes into play to solve this issue; \(i\) is defined as the square root of -1. Therefore, finding the square root of any negative number involves extracting \(i\) and treating the remaining positive part as a standard square root.

For example, to find \(\root{3}\root{-64}\), you'd rewrite this as \(\root{3}{8i}^2\) because \(\root{3}{-64} = \root{3}{64}*(-1) = 8i\). This helps us move forward with arithmetic operations using imaginary numbers while ensuring the equations remain manageable within the arithmetic rules set for complex numbers.

Exponents in Complex Numbers

Applying exponents in complex numbers might seem complex at first, but it follows the same principles as in regular algebra, with a twist due to the imaginary unit \(i\). Exponents of \(i\) exhibit a cyclic pattern: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\), then it repeats. In the context of our exercise, we calculate \((5\sqrt{3}i)^3\). Each component (the coefficient, the square root, and the imaginary unit) is raised to the power of 3 separately.

• The coefficient 5 becomes 125.
• The square root \(\sqrt{3}\) becomes \(\sqrt{27}\) or \(3\sqrt{3}\).
• The imaginary unit \(i\) taken to the third power is \(i^3 = -i\).

Combining these, we get \(-3750\sqrt{3}i\). Remembering the properties of exponents, along with the pattern of \(i\)'s powers, will greatly aid in manipulating and simplifying expressions with complex exponents.