In Exercises 55–78, use properties of rational exponents to simplify each expression. Assume that all variables represent positive numbers.3^¾ ⋅ 3^¼

Verified step by step guidance

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

Key Concepts

Rational Exponents

Rational exponents are a way to express roots and powers using fractions. For example, an exponent of 1/n indicates the nth root of a number, while m/n represents the nth root of the number raised to the mth power. Understanding this concept is crucial for simplifying expressions involving exponents, as it allows for the manipulation of roots and powers in a unified manner.

Properties of Exponents

The properties of exponents are rules that govern how to manipulate expressions with exponents. Key properties include the product of powers (a^m ⋅ a^n = a^(m+n)), the power of a power (a^m)^n = a^(m*n), and the power of a product (ab)^n = a^n ⋅ b^n. These properties are essential for simplifying expressions involving multiple bases and exponents.

Simplification of Expressions

Simplification involves rewriting an expression in a more manageable or concise form. In the context of exponents, this often means combining like terms, applying exponent rules, and reducing fractions. Mastering simplification techniques is vital for solving algebraic problems efficiently and accurately, especially when dealing with complex expressions.