Simplify each expression. See Example 1. (-8t³)(2t⁶)(-5t⁴)

Verified step by step guidance

1. Identify the coefficients and the variable parts in the expression \((-8t^3)(2t^6)(-5t^4)\) separately. The coefficients are \(-8\), \$2\(, and \)-5\(, and the variable parts are \)t^3\(, \)t^6\(, and \)t^4$.
2. Multiply the coefficients together: calculate \((-8) \times 2 \times (-5)\). Remember that multiplying two negative numbers results in a positive number.
3. Apply the product of powers property for the variable parts: when multiplying powers with the same base, add their exponents. So, add the exponents of \(t\): \$3 + 6 + 4$.
4. Write the simplified expression by combining the product of the coefficients and the variable with the new exponent: this will be in the form of a single coefficient multiplied by \(t\) raised to the sum of the exponents.
5. Double-check your signs and exponents to ensure the expression is fully simplified and correctly written.

Key Concepts

Multiplication of Coefficients

When multiplying expressions, multiply the numerical coefficients (constants) separately from the variables. For example, in (-8)(2)(-5), multiply the numbers to get the new coefficient before combining the variable parts.

Product of Powers Property

When multiplying variables with the same base, add their exponents. For instance, t^3 * t^6 * t^4 equals t^(3+6+4) = t^13. This property simplifies expressions with like bases.

Handling Negative Signs in Multiplication

Multiply the signs of the coefficients carefully: a negative times a positive is negative, and a negative times a negative is positive. This ensures the correct sign of the final product.