In Exercises 39–64, rationalize each denominator. 10 ---------- ⁵√16x²

Verified step by step guidance

1. Identify the denominator: $\sqrt[5]{16x^2}$.
2. To rationalize the denominator, multiply both the numerator and the denominator by $\sqrt[5]{{16}^4{x}^8}$ to make the denominator a perfect fifth power.
3. This gives: $\frac{10\cdot \sqrt[5]{{16}^4{x}^8}}{\sqrt[5]{16x^2}\cdot \sqrt[5]{{16}^4{x}^8}}$.
4. Simplify the denominator using the property $\sqrt[n]{a}\cdot \sqrt[n]{b}=\sqrt[n]{a\cdot b}$, resulting in $\sqrt[5]{(16x^2)\cdot ({16}^4{x}^8)}=\sqrt[5]{{16}^5{x}^{10}}$.
5. Since $\sqrt[5]{{16}^5{x}^{10}}=16x^2$, the expression becomes $\frac{10\cdot \sqrt[5]{{16}^4{x}^8}}{16x^2}$.

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator involves rewriting a fraction so that the denominator is a rational number. This is often necessary when the denominator contains roots or irrational numbers, as it simplifies calculations and makes the expression easier to work with. The process typically involves multiplying both the numerator and denominator by a suitable expression that eliminates the root in the denominator.

Radical Expressions

Radical expressions are mathematical expressions that include roots, such as square roots, cube roots, or higher-order roots. In the given problem, the expression involves a fifth root (⁵√), which indicates that we are dealing with a number raised to the power of one-fifth. Understanding how to manipulate and simplify these expressions is crucial for rationalizing denominators effectively.

Properties of Exponents

Properties of exponents are rules that govern how to manipulate expressions involving powers and roots. For instance, the property that states a^(m/n) = n√(a^m) helps in converting between radical and exponential forms. This understanding is essential when working with expressions like ⁵√(16x²), as it allows for easier simplification and rationalization of the denominator.