Problem 60 - College Algebra

Question

Evaluate each expression or indicate that the root is not a real number. $$\sqrt[4]{-81}$$

Step-by-Step Solution

Answer

The fourth root of -81 is not a real number.

Step 1: Expression

Evaluate each expression or indicate that the root is not a real number.
$$\sqrt[4]{-81}$$

Step 2: Radical Properties

\(\sqrt[n]{a/b}=\sqrt[n]{a}/\sqrt[n]{b}\), \(\sqrt[n]{a^m}=a^{m/n}\).

Step 3: Simplify

The fourth root of -81 is not a real number.

Key Concepts

Understanding Fourth Roots

The fourth root of a number is quite similar to square roots, but it extends the concept to multiplying a number by itself four times instead of just two. When you see an expression like \( \sqrt[4]{x} \), it means you're looking for a number \( y \) such that \( y^4 = x \).

Fourth roots are part of a broader group called nth roots, where you're finding a number that needs to be multiplied by itself 'n' times to get the original number.

For example, the fourth root of 16 is 2, because \( 2 \times 2 \times 2 \times 2 = 16 \). However, when it comes to negative numbers, things get tricky, as these can't really exist in the world of real numbers when talking about even roots, which we'll explore further below.

The Effect of Even Powers

Even powers play a critical role in determining the properties of real numbers. When you raise a number to an even power, it always results in a non-negative number.

For example:

• \( 2^2 = 4 \)
• \( (-2)^2 = 4 \)

In both cases, the results are positive. That's because multiplication makes the negative signs cancel out.

With fourth roots, you are essentially reversing this process. You are looking at a number that was squared twice, or raised to the fourth power, and trying to determine what that initial value could have been. However, if you start with a negative end result, like \( -81 \), there's no real number solution when dealing with even powers.

Negative Numbers Puzzle in Roots

Negative numbers present a unique challenge in root problems, especially with even roots. Since even powers eliminate negative signs, it's impossible for a negative number to be the result of a number raised to an even power.

This is why you cannot have a real number fourth root of \( -81 \). Any real number, when raised to the fourth power, will always be positive or zero, never negative.

It's important to remember this rule as it will guide you when you stumble upon problems asking for the square, fourth, or any even root of a negative number. The solution goes beyond real numbers into complex numbers where imaginary units come into play, which is a topic for deeper mathematical exploration.