Question

Find each root. See Example 3. $$ \sqrt[4]{\frac{81}{16}} $$

   Find each root. See Example 3.
$$
\sqrt[4]{\frac{81}{16}}
$$

Trigonometry

David I. Schneider;… 12th Edition

Chapter 0, Problem 46

Instant Answer

Solved by Expert Mukesh Devi

02/20/2023

Step 1

According to the rule of radicals, the nth root of a fraction can be written as the nth root of the numerator divided by the nth root of the denominator. So, we can rewrite the expression as $\frac{\sqrt[4]{81}}{\sqrt[4]{16}}$. Show more…

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Find each root. See Example 3. $$ \sqrt[4]{\frac{81}{16}} $$

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Key Concepts

Multiple Roots in Polynomial Equations

When solving equations that involve even roots, such as a fourth root, it is important to recognize that there are multiple solutions, including both positive and negative real values, as well as complex solutions when necessary. This concept extends to finding all nth roots of a number and often involves representing the solutions in polar form or using the concept of roots of unity.

Radical Expressions and Nth Roots

This concept involves understanding that an nth root of a number refers to a value which, when raised to the nth power, returns the original number. It forms a basic part of algebra and serves as the inverse operation to exponentiation. In many cases, the process includes identifying the principal (typically real and nonnegative) root, as well as the additional roots when dealing with even-degree radicals.

Rational Exponents and Exponent Laws

Expressing roots in terms of exponents is a common technique used to simplify and manipulate radical expressions. A radical such as the fourth root can be rewritten using a rational exponent (for example, a^(1/4)), allowing the use of exponent laws. This is particularly useful when the radicand is a fraction or product of numbers raised to powers.

Simplifying Radicals Using Perfect Powers

Identifying perfect power factors in the numerator and denominator of a fraction under a radical is an important skill. It allows one to simplify the radical by rewriting the numbers as bases raised to powers that match the radical’s index. For example, if the numerator and denominator are both perfect fourth powers, the radical can be simplified directly by taking the fourth root of each.

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