In Exercises 55–78, use properties of rational exponents to simplify each expression. Assume that all variables represent positive numbers.(25x⁴y⁶)^½

Verified step by step guidance

1. Identify the expression: \((25x^4y^6)^{\frac{1}{2}}\).
2. Apply the property of exponents: \((a^m)^n = a^{m \cdot n}\).
3. Distribute the exponent \(\frac{1}{2}\) to each term inside the parentheses: \((25^{\frac{1}{2}})(x^{4 \cdot \frac{1}{2}})(y^{6 \cdot \frac{1}{2}})\).
4. Simplify each term: \(25^{\frac{1}{2}} = \sqrt{25}\), \(x^{4 \cdot \frac{1}{2}} = x^2\), \(y^{6 \cdot \frac{1}{2}} = y^3\).
5. Combine the simplified terms: \(\sqrt{25} \cdot x^2 \cdot y^3\).

Key Concepts

Rational Exponents

Rational exponents are a way to express roots using fractional powers. For example, an exponent of 1/2 indicates the square root, while 1/3 represents the cube root. This notation allows for easier manipulation of expressions, especially when combined with other algebraic operations.

Properties of Exponents

The properties of exponents include rules such as the product of powers, power of a power, and power of a product. These rules help simplify expressions involving exponents by allowing us to combine or separate terms effectively. For instance, (a^m)^n = a^(m*n) is crucial for simplifying expressions with rational exponents.

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. In the context of rational exponents, it means rewriting expressions in a way that makes them easier to understand or compute, such as converting back to radical form if necessary.