Simplify each expression. See Example 1. (-8t^3)(2t^6)(-5t^4)

Verified step by step guidance

1. First, multiply the coefficients: $(-8)\times 2\times (-5)$.
2. Next, apply the product of powers property to the variables: $t^3\times t^6\times t^4$.
3. Combine the coefficients from step 1.
4. Add the exponents of the like bases from step 2: $3+6+4$.
5. Write the final expression by combining the result from the coefficients and the sum of the exponents.

Key Concepts

Multiplication of Monomials

Multiplying monomials involves multiplying their coefficients and adding their exponents when the bases are the same. For example, in the expression (-8t^3)(2t^6), you multiply -8 and 2 to get -16, and then add the exponents of t, resulting in t^(3+6) = t^9.

Properties of Exponents

The properties of exponents are rules that govern how to handle expressions involving powers. Key rules include the product of powers (a^m * a^n = a^(m+n)) and the power of a product ((ab)^n = a^n * b^n), which are essential for simplifying expressions with the same base.

Sign Rules in Multiplication

When multiplying numbers, the sign of the product depends on the signs of the factors. Specifically, the product of two negative numbers is positive, while the product of a positive and a negative number is negative. Understanding these rules is crucial for determining the sign of the final result in expressions like (-8)(2)(-5).