In Exercises 55–78, use properties of rational exponents to simplify each expression. Assume that all variables represent positive numbers.3^½ ⋅ 3^¾3^¼

Verified step by step guidance

1. Identify the expression: \(3^{1/2} \cdot 3^{3/4} \cdot 3^{1/4}\).
2. Use the property of exponents that states \(a^m \cdot a^n = a^{m+n}\) to combine the exponents.
3. Add the exponents: \(\frac{1}{2} + \frac{3}{4} + \frac{1}{4}\).
4. Convert all fractions to have a common denominator, which is 4 in this case.
5. Simplify the expression by adding the fractions: \(\frac{2}{4} + \frac{3}{4} + \frac{1}{4}\).

Key Concepts

Properties of Exponents

The properties of exponents are rules that govern how to manipulate expressions involving powers. 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. Understanding these properties is essential for simplifying expressions with exponents.

Rational Exponents

Rational exponents are exponents that are expressed as fractions. For example, a^(m/n) represents the n-th root of a raised to the m-th power. This concept allows for the conversion between radical expressions and exponential forms, facilitating easier manipulation and simplification of expressions involving roots and powers.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form while maintaining equivalence. This process often includes combining like terms, applying exponent rules, and reducing fractions. Mastery of simplification techniques is crucial for solving algebraic problems efficiently and accurately.