In Exercises 39–54, rewrite each expression with a positive rational exponent. Simplify, if possible. (2xy)^-7/10

Verified step by step guidance

1. Identify the expression with a negative exponent: \((2xy)^{-\frac{7}{10}}\).
2. Recall that a negative exponent indicates the reciprocal of the base raised to the positive of that exponent.
3. Rewrite the expression as the reciprocal: \(\frac{1}{(2xy)^{\frac{7}{10}}}\).
4. Recognize that the expression is now written with a positive rational exponent.
5. Simplify further if possible, but in this case, the expression is already simplified.

Key Concepts

Negative Exponents

A negative exponent indicates that the base should be taken as the reciprocal. For example, a term like a^-n can be rewritten as 1/a^n. This concept is essential for transforming expressions with negative exponents into a more manageable form, particularly when simplifying algebraic expressions.

Rational Exponents

Rational exponents represent roots and powers simultaneously. An exponent in the form of m/n indicates the n-th root of the base raised to the m-th power. Understanding this concept allows for the rewriting of expressions in a way that combines both exponentiation and root extraction, facilitating simplification.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, which often includes combining like terms, reducing fractions, and applying exponent rules. This process is crucial for making complex expressions easier to work with and understand, especially when dealing with exponents and roots.