Find each root. See Example 3.-∜16

Verified step by step guidance

1. Recognize that the problem is asking for the fourth root of 16, which can be expressed as \(-\sqrt[4]{16}\).
2. Recall that finding the fourth root of a number is equivalent to raising that number to the power of \(\frac{1}{4}\).
3. Express the problem in exponential form: \(-16^{\frac{1}{4}}\).
4. Determine the positive fourth root of 16, which is the number that when raised to the fourth power equals 16.
5. Apply the negative sign to the positive fourth root to find the final result.

Key Concepts

Roots and Radicals

Roots and radicals are mathematical operations that involve finding a number that, when raised to a certain power, yields a given value. The notation ∜16 indicates the fourth root of 16, which means finding a number that, when multiplied by itself four times, equals 16. Understanding how to manipulate and simplify radical expressions is essential for solving problems involving roots.

Even and Odd Roots

Even and odd roots have different properties. An even root, such as the square root or fourth root, can yield both positive and negative results, as both can satisfy the equation when raised to an even power. In contrast, odd roots, like the cube root, yield a single real result, which can be negative. Recognizing these distinctions is crucial when determining the roots of a number.

Simplifying Radicals

Simplifying radicals involves expressing a radical in its simplest form, which often includes factoring out perfect squares or higher powers. For example, the fourth root of 16 can be simplified by recognizing that 16 is 2 raised to the fourth power, leading to a straightforward solution. Mastery of this concept allows for more efficient calculations and clearer understanding of radical expressions.