In Exercises 55–78, use properties of rational exponents to simplify each expression. Assume that all variables represent positive numbers.(3y^¼)³y^1/12

Verified step by step guidance

1. Step 1: Start by simplifying the expression \((3y^{1/4})^3\). Use the property of exponents \((a^m)^n = a^{m \cdot n}\) to simplify the expression.
2. Step 2: Apply the exponent rule to \((3y^{1/4})^3\), which gives \(3^3 \cdot (y^{1/4})^3\).
3. Step 3: Simplify \(3^3\) to get 27, and use the exponent rule \((y^{1/4})^3 = y^{(1/4) \cdot 3}\).
4. Step 4: Calculate \((1/4) \cdot 3\) to simplify \(y^{3/4}\).
5. Step 5: Combine the simplified expression \(27y^{3/4}\) with \(y^{1/12}\) using the property \(a^m \cdot a^n = a^{m+n}\) to get the final simplified form.

Key Concepts

Rational Exponents

Rational exponents are a way to express roots using fractional powers. For example, an exponent of 1/n indicates the n-th root of a number. This concept allows for the simplification of expressions involving roots and powers, making it easier to manipulate algebraic expressions.

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 help in simplifying expressions by allowing us to combine or separate bases and exponents systematically. Understanding these properties is essential for effectively simplifying expressions with rational exponents.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, often by combining like terms and applying exponent rules. This process is crucial in algebra as it makes expressions easier to work with and understand. In the context of rational exponents, simplification often requires careful application of exponent properties to achieve a more manageable form.