Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent nonzero real numbers. See Examples 5 and 6. (5a^-1)⁴(a²)^-3

Verified step by step guidance

1. Start by applying the power of a power rule to each term inside the parentheses. For \((5a^{-1})^4\), raise both 5 and \(a^{-1}\) to the 4th power: \((5)^4\) and \((a^{-1})^4\).
2. Similarly, for \((a^2)^{-3}\), apply the power of a power rule by multiplying the exponents: \(a^{2 imes (-3)}\).
3. Rewrite the expression using the results from the previous step: \$5^4 imes a^{-4} imes a^{-6}$.
4. Combine the powers of \(a\) by adding the exponents since the bases are the same: \(a^{-4} imes a^{-6} = a^{-4 + (-6)}\).
5. Rewrite the expression without negative exponents by using the rule \(a^{-n} = \frac{1}{a^n}\), and simplify the expression accordingly.

Key Concepts

Laws of Exponents

The laws of exponents govern how to simplify expressions involving powers. Key rules include the power of a power rule ((a^m)^n = a^(m*n)) and the product of powers rule (a^m * a^n = a^(m+n)). These rules allow combining and simplifying exponential expressions efficiently.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the positive exponent, such that a^(-n) = 1/a^n. To write answers without negative exponents, rewrite terms with negative powers as fractions with positive exponents in the denominator.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves applying exponent rules and combining like terms to write the expression in its simplest form. This process often requires careful handling of coefficients, variables, and exponents to ensure clarity and correctness.