Problem 60 - Algebra and Trigonometry

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 is not a real number.

Step 1: Analyzing the Radicand

Analyze the radicand, which is -81. It's a negative number. Then, note the index of the root, which is 4, an even number.

Step 2: Evaluating the Root

Since the index of the root is an even number, \( \sqrt[4]{-81} \) cannot be a real number. A rule of basic algebra is that you cannot extract an even root from a negative number in the set of real numbers.

Step 3: Conclusion

Since the rule does not allow extracting an even root from a negative number in the realm of real numbers, we deem that \( \sqrt[4]{-81} \) is not a real number.

Key Concepts

Even Roots

Even roots mean the index of the root is an even number. When we talk about taking even roots, we are referring to expressions like square roots, fourth roots, sixth roots, and so on. If we look specifically at the fourth root as in our exercise, it's essentially asking which number multiplied by itself four times results in the radicand.

A special characteristic of even roots is that they require positive numbers under the radicand to produce a real number outcome.

• If the number under the root (the radicand) is negative, the result will not be a real number because no real number exists that can multiply by itself an even number of times to yield a negative product.
• On the contrary, if the radicand is positive, the even root calculation yields a real number.

This is why expressions like \(\sqrt[4]{-81}\) do not result in a real number.

Negative Numbers

Negative numbers are less than zero and are usually represented with a minus sign. They are commonly found on the left half of the number line. In mathematics, they behave differently than positive numbers, especially when it comes to roots and exponents.

• For odd roots, such as cube roots, a negative number can still yield a real number. For example, \(\sqrt[3]{-27} = -3\).
• However, for even roots, having a negative number as the radicand means the result is not a real number, because multiplying a real number by itself an even number of times never results in a negative product.

This distinction is essential in solving algebraic expressions involving roots.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They form the backbone of algebra and allow us to describe mathematical situations in a concise way.

When dealing with roots in algebraic expressions, especially even roots, it's crucial to consider whether the radicand is positive or negative. This decision impacts the result:

• For even roots such as \(\sqrt[4]{x}\), if \(x\) is negative, the expression is not a real number.
• Algebraic manipulations must follow these principles to ensure accurate and valid outcomes in mathematical procedures.

These concepts are critical in understanding why certain expressions do not yield real numbers and ensuring correct calculations in mathematics.