Simplify each expression. Write answers without negative exponents. Assume all variables represent nonzero real numbers. -4r^-2(r⁴)²

Verified step by step guidance

1. Start by rewriting the expression clearly: \(-4r^{-2}(r^{4})^{2}\).
2. Apply the power of a power rule to the term \((r^{4})^{2}\), which means multiply the exponents: \(r^{4 \times 2} = r^{8}\).
3. Now the expression becomes \(-4r^{-2} \times r^{8}\).
4. Use the product of powers rule to combine the terms with the same base \(r\): add the exponents \(-2 + 8\) to get \(r^{6}\).
5. Rewrite the expression as \(-4r^{6}\), which has no negative exponents and is fully simplified.

Key Concepts

Laws of Exponents

The laws of exponents govern how to simplify expressions involving powers, such as multiplying powers with the same base by adding exponents, and raising a power to another power by multiplying exponents. Understanding these rules is essential to simplify expressions like (-4r^{-2})(r^{4})^{2} correctly.

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. For example, r^{-2} equals 1/r^{2}. Converting negative exponents to positive ones is necessary to write the final answer without negative exponents, as required by the problem.

Simplification of Algebraic Expressions

Simplification involves combining like terms and applying exponent rules to rewrite expressions in their simplest form. This includes removing parentheses by applying exponent rules and ensuring the expression is expressed clearly without negative exponents, making it easier to interpret and use.