In Exercises 85–116, simplify each exponential expression.(2a⁵)(-3a⁻⁷)

Verified step by step guidance

1. Identify the coefficients and the variables in the expression: \((2a^5)(-3a^{-7})\).
2. Multiply the coefficients: \(2\) and \(-3\).
3. Apply the product of powers property for the variable \(a\): \(a^5 \cdot a^{-7}\).
4. Use the property \(a^m \cdot a^n = a^{m+n}\) to combine the exponents: \(a^{5 + (-7)}\).
5. Simplify the expression by calculating the new exponent for \(a\).

Key Concepts

Exponential Rules

Exponential rules govern how to simplify expressions involving powers. Key rules include the product of powers rule, which states that when multiplying like bases, you add the exponents, and the power of a power rule, which states that when raising a power to another power, you multiply the exponents. Understanding these rules is essential for simplifying expressions like (2a⁵)(-3a⁻⁷).

Combining Like Terms

Combining like terms involves simplifying expressions by adding or subtracting coefficients of terms that have the same variable raised to the same power. In the expression (2a⁵)(-3a⁻⁷), recognizing that both terms involve the variable 'a' allows for the combination of their coefficients after applying the exponential rules, leading to a more simplified form.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the absolute value of the exponent. For example, a⁻ⁿ = 1/aⁿ. This concept is crucial when simplifying expressions like (2a⁵)(-3a⁻⁷), as it allows for the transformation of negative exponents into positive ones, facilitating further simplification of the expression.