In Exercises 53–58, simplify each expression using the power rule.(7⁻⁴)⁻⁵

Verified step by step guidance

1. Identify the power rule for exponents: \((a^m)^n = a^{m \cdot n}\).
2. Apply the power rule to the expression \((7^{-4})^{-5}\).
3. Multiply the exponents: \(-4 \times -5\).
4. Simplify the multiplication of the exponents to find the new exponent.
5. Rewrite the expression with the simplified exponent: \(7^{20}\).

Key Concepts

Power Rule

The power rule states that when raising a power to another power, you multiply the exponents. Mathematically, this is expressed as (a^m)^n = a^(m*n). This rule is essential for simplifying expressions involving exponents, allowing for easier manipulation of terms.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the opposite positive exponent. For example, a^(-n) = 1/(a^n). Understanding how to work with negative exponents is crucial for simplifying expressions that involve them, as it transforms them into a more manageable form.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, which often includes combining like terms, applying exponent rules, and eliminating unnecessary components. This process is vital in algebra as it helps clarify the expression and makes further calculations easier.