In Exercises 1–20, use radical notation to rewrite each expression. Simplify, if possible. (-27)^⅓

Verified step by step guidance

1. Recognize that the expression $(-27{)}^{\frac{1}{3}}$ represents the cube root of $-27$.
2. Rewrite the expression using radical notation: $\sqrt[3]{-27}$.
3. Recall that the cube root of a negative number is also negative, because multiplying three negative numbers results in a negative product.
4. Identify that $-27$ can be expressed as $(-3)\times (-3)\times (-3)$, which means $-3$ is the cube root of $-27$.
5. Conclude that $\sqrt[3]{-27}=-3$.

Key Concepts

Radical Notation

Radical notation is a mathematical notation used to represent roots of numbers. The expression √x denotes the square root of x, while x^(1/n) represents the nth root of x. Understanding how to convert between radical and exponential forms is essential for simplifying expressions involving roots.

Cube Root

The cube root of a number x, denoted as x^(1/3) or ∛x, is the value that, when multiplied by itself three times, gives x. For negative numbers, the cube root is also negative, which is important when dealing with expressions like (-27)^(1/3). Recognizing this property helps in simplifying radical expressions correctly.

Simplification of Radicals

Simplifying radicals involves reducing the expression to its simplest form, which often includes factoring out perfect squares or cubes. For example, when simplifying (-27)^(1/3), one should recognize that -27 can be expressed as (-3)^3, leading to a straightforward simplification. Mastery of this concept is crucial for effectively rewriting and simplifying radical expressions.