In Exercises 55–78, use properties of rational exponents to simplify each expression. Assume that all variables represent positive numbers. x^⅘ x^⅕

Verified step by step guidance

1. Identify the expression: \( x^{\frac{4}{5}} \times x^{\frac{1}{5}} \).
2. Recall the property of exponents: \( a^m \times a^n = a^{m+n} \).
3. Apply the property to combine the exponents: \( x^{\frac{4}{5} + \frac{1}{5}} \).
4. Add the exponents: \( \frac{4}{5} + \frac{1}{5} = \frac{5}{5} \).
5. Simplify the expression: \( x^{\frac{5}{5}} = x^1 \).

Key Concepts

Rational Exponents

Rational exponents are a way to express roots using fractional powers. For example, the expression x^(m/n) represents the n-th root of x raised to the m-th power. Understanding how to manipulate these exponents is crucial for simplifying expressions involving them.

Properties of Exponents

The properties of exponents include rules such as the product of powers, power of a power, and quotient of powers. These rules allow us to combine and simplify expressions with exponents effectively. For instance, when multiplying like bases, we add the exponents, which is essential for simplifying the given expression.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form while maintaining equivalence. This process often includes combining like terms, applying exponent rules, and reducing fractions. In the context of rational exponents, it requires careful application of the properties of exponents to achieve a more manageable expression.