Simplify each exponential expression in Exercises 23–64. x⁻²y

Verified step by step guidance

1. Identify the expression to simplify: \(x^{-2}y\). Notice that \(x\) has a negative exponent.
2. Recall the rule for negative exponents: \(a^{-n} = \frac{1}{a^n}\), where \(a \neq 0\) and \(n\) is a positive integer.
3. Apply the negative exponent rule to \(x^{-2}\), rewriting it as \(\frac{1}{x^2}\).
4. Rewrite the entire expression by substituting \(x^{-2}\) with \(\frac{1}{x^2}\), so the expression becomes \(\frac{y}{x^2}\).
5. Check if the expression can be simplified further. Since \(y\) and \(x^2\) have no common factors, the simplified form is \(\frac{y}{x^2}\).

Key Concepts

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. For example, x^(-2) equals 1 divided by x squared, or 1/x^2. Understanding this helps simplify expressions with negative powers.

Product of Powers

When multiplying expressions with the same base, add their exponents. For instance, x^a * x^b = x^(a+b). This rule is essential when simplifying expressions involving multiple powers of the same variable.

Variables and Exponents

Each variable in an expression is treated independently with its exponent. For example, in x^(-2)y, x and y are separate factors, so the exponent applies only to x. Recognizing this prevents incorrect combining of terms.