Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent nonzero real numbers. See Examples 5 and 6. 4a⁵(a^-1)³/(a^-2)^-2

Verified step by step guidance

1. Start by rewriting the expression clearly: \(\frac{4a^5 (a^{-1})^3}{(a^{-2})^{-2}}\).
2. Apply the power of a power rule to each term with exponents raised to another exponent: \((a^{-1})^3 = a^{-1 \times 3} = a^{-3}\) and \((a^{-2})^{-2} = a^{-2 \times -2} = a^4\).
3. Substitute these back into the expression to get: \(\frac{4a^5 \cdot a^{-3}}{a^4}\).
4. Combine the terms in the numerator by adding exponents of like bases: \(a^5 \cdot a^{-3} = a^{5 + (-3)} = a^2\).
5. Rewrite the expression as \(\frac{4a^2}{a^4}\) and then subtract exponents in the division: \(a^{2 - 4} = a^{-2}\). Finally, rewrite to eliminate the negative exponent by expressing it 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 help simplify complex expressions systematically.

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 allows rewriting expressions without negative exponents, which is often required for final answers in algebra.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves combining like terms and applying exponent rules to rewrite expressions in simpler forms. This process includes distributing exponents over products, handling parentheses, and ensuring the final expression meets given conditions, such as no negative exponents.