Simplify each expression. Write answers without negative exponents. Assume all variables represent nonzero real numbers. 4a⁵(a^-1)³/(a^-2)^-2

Verified step by step guidance

1. Start by rewriting the expression clearly: \(\frac{4a^{5} \left(a^{-1}\right)^{3}}{\left(a^{-2}\right)^{-2}}\).
2. Apply the power of a power rule \(\left(x^{m}\right)^{n} = x^{m \cdot n}\) to simplify each term with exponents raised to another exponent: \(\left(a^{-1}\right)^{3} = a^{-3}\) and \(\left(a^{-2}\right)^{-2} = a^{4}\).
3. Substitute these back into the expression to get \(\frac{4a^{5} \cdot a^{-3}}{a^{4}}\).
4. Use the product rule for exponents \(a^{m} \cdot a^{n} = a^{m+n}\) to combine the terms in the numerator: \(a^{5} \cdot a^{-3} = a^{5 + (-3)} = a^{2}\).
5. Now the expression is \(\frac{4a^{2}}{a^{4}}\). Use the quotient rule for exponents \(\frac{a^{m}}{a^{n}} = a^{m-n}\) to simplify: \(a^{2 - 4} = a^{-2}\). Finally, rewrite to eliminate the negative exponent as \(\frac{4}{a^{2}}\).

Key Concepts

Laws of Exponents

The laws of exponents govern how to simplify expressions involving powers. Key rules include multiplying powers with the same base by adding exponents, raising a power to another power by multiplying exponents, and dividing powers by subtracting exponents. These rules allow for systematic simplification of expressions like the one given.

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. For example, a^{-n} = 1/a^n. Understanding this helps rewrite expressions without negative exponents, as required in the problem, by converting them into positive exponents in the numerator or denominator.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves combining like terms and applying exponent rules to write the expression in its simplest form. This includes carefully handling parentheses, exponents, and ensuring the final answer meets given conditions, such as no negative exponents and assuming variables are nonzero to avoid undefined expressions.