In Exercises 55–78, use properties of rational exponents to simplify each expression. Assume that all variables represent positive numbers.x^½ ⋅ x^⅓

Verified step by step guidance

1. Identify the expression: \( x^{\frac{1}{2}} \cdot x^{\frac{1}{3}} \).
2. Recall the property of exponents: \( a^m \cdot a^n = a^{m+n} \).
3. Apply the property to combine the exponents: \( x^{\frac{1}{2} + \frac{1}{3}} \).
4. Find a common denominator to add the fractions: \( \frac{1}{2} = \frac{3}{6} \) and \( \frac{1}{3} = \frac{2}{6} \).
5. Add the fractions: \( \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \), so the expression simplifies to \( x^{\frac{5}{6}} \).

Key Concepts

Rational Exponents

Rational exponents are a way to express roots using fractional powers. For example, x^(1/n) represents the n-th root of x. Understanding how to manipulate these exponents is crucial for simplifying expressions involving them, as they allow us to rewrite roots in a more manageable form.

Properties of Exponents

The properties of exponents, such as the product of powers property, state that when multiplying two expressions with the same base, you add their exponents. For instance, x^a ⋅ x^b = x^(a+b). This property is essential for simplifying expressions with rational exponents, as it enables the combination of terms efficiently.

Simplification of Expressions

Simplification involves rewriting an expression in a more concise or manageable form. In the context of rational exponents, this often means combining terms, reducing fractions, or eliminating radicals. Mastering simplification techniques is vital for solving algebraic problems and understanding the underlying relationships between variables.