Problem 60 - College Algebra Essentials

Question

Evaluate each expression in Exercises \(55-66,\) or indicate that the root is not a real number. $$\sqrt[4]{-81}$$

Step-by-Step Solution

Answer

The fourth root of -81 in complex numbers is \(3i\).

Step 1: Identify the negative sign

First, we need to note that the expression involves a negative number under the root sign. In reality, roots of negative numbers are not defined. However, in the field of complex numbers, these roots are represented using 'i', the imaginary unit, where \(i=\sqrt{-1}\). Thus, we rewrite the expression \(\sqrt[4]{-81}\) as \(i\sqrt[4]{81}\).

Step 2: Compute the fourth root of 81

Now, we can compute the value of the fourth root of 81. We can simplify \(\sqrt[4]{81}\) by searching for the number that will give 81 when raised to the power of 4. This number is 3, since \(3^4=81\). So, \(\sqrt[4]{81}=3\). Substituting this into the previous expression gives us \(i*3\).

Step 3: Rewrite and simplify the final answer

Finally, we can simplify the expression further by writing \(i*3\) as \(3i\). This represents the fourth root of -81 in complex numbers.

Key Concepts

Imaginary Unit

The imaginary unit, denoted by the symbol \(i\), is a special mathematical construct used to handle the square roots of negative numbers. In everyday mathematics, you might remember from class that no real number can satisfy \(x^2 = -1\), simply because any real number squared is non-negative.

The introduction of the imaginary unit allows us to circumvent this limitation. In the world of complex numbers, \(i\) is defined as \(\sqrt{-1}\). This innovative concept opens up a whole new field called complex numbers.

• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\)

The above identities are particularly useful when working with powers of \(i\). When solving equations with negative numbers under a root sign, the imaginary unit \(i\) turns an unmanageable situation into an elegant solution.

Fourth Root

Finding the fourth root of a number means looking for a number that, when multiplied by itself four times, gives the original number. Mathematically, the fourth root of a number \(x\) is represented as \(\sqrt[4]{x}\).

In the example of \(\sqrt[4]{81}\), we seek a number \(y\) such that \(y^4 = 81\). By trial or by recognizing patterns from powers, we find that 3 is the appropriate number because \(3^4 = 81\).

• \(\sqrt[4]{81} = 3\)
• \(x = x^{1/4}\)

Understanding roots beyond the basic square root extends your ability to solve complex number problems and assist in simplifying expressions easily.

Real Numbers

Real numbers form the broad collection of numbers that can be found on the number line. These numbers include both rational numbers (like 1/2 or 2) and irrational numbers (like \(\pi\) or \(\sqrt{2}\)).

They are what we typically use in basic arithmetic. Real numbers do not include the square roots of negative numbers, as these do not have a solution within the real number system without involving the imaginary unit.

• Integer examples: -3, 0, 7
• Decimal examples: 0.5, -2.7

If you attempt to take an even root (such as a square root or fourth root) of a negative number within the realm of real numbers, it yields no solution. This is where the unique world of complex numbers comes into play.

Negative Numbers in Roots

Roots of negative numbers pose interesting challenges in traditional arithmetic since real numbers alone can't handle them elegantly. If you try to find even roots like the square root or the fourth root of a negative number without extending into complex numbers, you'll find it doesn't work.

Negative numbers have a transformative relationship with the imaginary unit \(i\). For instance, considering \(\sqrt[4]{-81}\), you rewrite this problem with the help of \(i\), explicitly noted as \(i\sqrt[4]{81}\).

• \(\sqrt[4]{-1} = i\)
• Even roots of negatives involve \(i\)

In this case, the real fourth root of 81 is calculated as shown previously, and combined with \(i\), the solution becomes complete in the complex plane, yielding \(3i\). This highlights how complex numbers expand mathematical possibilities.