In Exercises 65–74, simplify each radical expression and then rationalize the denominator.15------------³√-27x⁴y¹¹

Verified step by step guidance

1. Identify the cube root in the denominator: \( \sqrt[3]{-27x^4y^{11}} \).
2. Simplify the cube root: \( \sqrt[3]{-27} = -3 \), \( \sqrt[3]{x^4} = x^{4/3} \), and \( \sqrt[3]{y^{11}} = y^{11/3} \).
3. Rewrite the expression with the simplified denominator: \( \frac{15}{-3x^{4/3}y^{11/3}} \).
4. Rationalize the denominator by multiplying the numerator and the denominator by \( x^{2/3}y^{1/3} \) to make the exponents in the denominator whole numbers.
5. Simplify the expression after rationalization to ensure the denominator is free of radicals.

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. In this case, we are dealing with a cube root, which is the inverse operation of raising a number to the third power. Understanding how to simplify radical expressions is crucial, as it involves breaking down the expression into its prime factors and applying the properties of exponents.

Rationalizing the Denominator

Rationalizing the denominator is the process of eliminating any radicals from the denominator of a fraction. This is typically done by multiplying both the numerator and the denominator by a suitable expression that will result in a rational number in the denominator. This step is important for presenting the final answer in a standard form that is easier to interpret.

Properties of Exponents

The properties of exponents govern how to manipulate expressions involving powers. Key rules include the product of powers, quotient of powers, and power of a power. These rules are essential when simplifying radical expressions, as they help in rewriting roots in terms of fractional exponents, which can then be simplified further.