Problem 477 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(\sqrt[3]{-8}\) (b) \(\sqrt[4]{-16}\)

Step-by-Step Solution

Answer

a) -2, b) Not a real number

Step 1: Simplify \(\sqrt[3]{-8}\)

Since the cube root of a negative number is also negative, we find the cube root of 8 and then apply the negative sign. The cube root of 8 is 2 because \(2^3 = 8\). Therefore, \(\sqrt[3]{-8} = -2\).

Step 2: Simplify \(\sqrt[4]{-16}\)

The fourth root of a negative number is not a real number, because there is no real number that, when raised to the fourth power, equals a negative number. Thus, \(\sqrt[4]{-16}\) is not a real number.

Key Concepts

Cube Roots

Cube roots are the opposite of cubing a number. When you need to find the cube root, you are looking for a number that, when multiplied by itself three times (cubed), gives you the original number. For example, \(\root[3]{27} = 3\) because \({3}^3 = 27\). Cube roots can handle negative numbers. So, \(\root[3]{-8} = -2\) because \(-2 \times -2 \times -2 = -8\). When simplifying cube roots, remember these points:

• Cube roots of positive numbers are positive.
• Cube roots of negative numbers are negative.

Knowing this can help us simplify expressions involving cube roots quickly and accurately.

Fourth Roots

Fourth roots demand a bit more care. To find the fourth root, you need a number that multiplies by itself four times to give the original number. For instance, \(\root[4]{16} = 2\), because \({2}^4 = 16\). Notice how even the fourth powers of both positive and negative numbers result in positive values.
Therefore, for any negative number, there isn't a real number that satisfies the equation. For example, \(\root[4]{-16}\) does not yield a real result because no real number times itself four times makes a negative product. This is why \(\root[4]{-16}\) is not a real number.

Negative Numbers

When dealing with roots of negative numbers, the type of root is important:

• Cube Roots: The cube root of a negative number is a negative number.
• Fourth Roots: The fourth root of a negative number is not a real number.

In general, odd roots (like cube roots) can handle negatives because multiplying an odd number of negative values results in a negative value. Even roots (like square roots or fourth roots), however, do not yield real numbers with negative radicands because multiplying an even number of negative values results in a positive value.
Always remember to check the type of root you are dealing with, and whether the original number is positive or negative.