In Exercises 1–20, use radical notation to rewrite each expression. Simplify, if possible.(xy)^4/7

Verified step by step guidance

1. Identify the expression given: \((xy)^{\frac{4}{7}}\).
2. Recognize that the expression is in the form of a rational exponent, where the numerator is the power and the denominator is the root.
3. Rewrite the expression using radical notation: \(\sqrt[7]{(xy)^4}\).
4. Simplify the expression inside the radical, if possible. In this case, \((xy)^4\) can be expanded to \(x^4y^4\).
5. The expression in radical form is \(\sqrt[7]{x^4y^4}\).

Key Concepts

Radical Notation

Radical notation is a way to express roots of numbers or expressions using the radical symbol (√). For example, the square root of a number 'a' is written as √a. In algebra, radical notation can also represent fractional exponents, where the denominator indicates the root and the numerator indicates the power. Understanding this notation is essential for rewriting expressions involving roots.

Exponents and Fractional Exponents

Exponents represent repeated multiplication of a base number. A fractional exponent, such as 4/7, indicates both a root and a power: the denominator (7) signifies the root, while the numerator (4) indicates the exponent applied after taking the root. For instance, a^(m/n) can be rewritten as the nth root of a raised to the m power, which is crucial for simplifying expressions involving exponents.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, making them easier to work with. This can include combining like terms, reducing fractions, and applying the laws of exponents and radicals. In the context of the given expression, simplifying may involve rewriting it in a more manageable form using radical notation and ensuring that all components are expressed clearly and concisely.