Question

Find each root. See Example 3. $$ \sqrt[3]{\frac{64}{27}} $$

   Find each root. See Example 3.
$$
\sqrt[3]{\frac{64}{27}}
$$

Trigonometry

David I. Schneider;… 12th Edition

Chapter 0, Problem 45

Instant Answer

Solved by Expert Mukesh Devi

02/20/2023

Step 1

According to the quotient rule of radicals, the root of a quotient is equal to the quotient of the roots. So, we can rewrite the expression as $\frac{\sqrt[3]{64}}{\sqrt[3]{27}}$. Show more…

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Find each root. See Example 3. $$ \sqrt[3]{\frac{64}{27}} $$

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Key Concepts

Radical Expressions and Rational Exponents

This concept involves understanding how radicals can be expressed as rational exponents, which allows for the manipulation and simplification of expressions. For example, the cube root of a number can be written as that number raised to the 1/3 power, letting you apply the usual rules of exponents to simplify or compute the expression.

Complex Numbers and Roots

When extracting roots beyond the principal (or real) root, especially for non-negative numbers, it is important to consider that there may be multiple complex roots. This concept covers how every n-th root equation has n distinct solutions in the complex plane, even if some of these solutions are not real.

De Moivre's Theorem

De Moivre's Theorem provides a method for finding roots of complex numbers when they are expressed in polar form. It states that for a complex number expressed as r*(cos ? + i*sin ?), its n-th roots can be found by taking the n-th root of r and dividing the angle ? (plus multiples of 360° or 2?) by n, ensuring all distinct roots are obtained.

Roots of Unity

This concept refers to the n distinct complex numbers that, when raised to a given power, equal one. In practice, when finding the n-th roots of any number, the additional solutions are generated by multiplying the principal root with the various roots of unity, allowing all possible roots in the complex plane to be identified.

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