Simplify each expression. Write answers without negative exponents. Assume all variables represent nonzero real numbers. (5a^-1)⁴(a²)^-3

Verified step by step guidance

1. Start by rewriting the expression clearly: \(\left(5a^{-1}\right)^4 \left(a^2\right)^{-3}\).
2. Apply the power of a power rule, which states that \((x^m)^n = x^{m \cdot n}\), to each part separately: \(5^4 \cdot (a^{-1})^4 \cdot a^{2 \cdot (-3)}\).
3. Simplify the exponents: \(5^4 \cdot a^{-4} \cdot a^{-6}\).
4. Combine the terms with the same base \(a\) by adding their exponents: \(a^{-4 + (-6)} = a^{-10}\).
5. Rewrite the expression without negative exponents by using the rule \(x^{-m} = \frac{1}{x^m}\), resulting in \(5^4 \cdot \frac{1}{a^{10}}\).

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, raising a power to another power by multiplying exponents, and handling negative exponents by rewriting them as reciprocals.

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. For example, a⁻¹ equals 1/a. Simplifying expressions often involves rewriting negative exponents as positive by moving factors between numerator and denominator.

Simplification of Algebraic Expressions

Simplification involves combining like terms and applying exponent rules to write expressions in their simplest form. This includes eliminating negative exponents, reducing powers, and ensuring the final answer is clear and concise, as required in algebra problems.