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

Verified step by step guidance

1. Identify the coefficients (numerical parts) and the variable parts separately in the expression \((-8t^{3})(2t^{6})(-5t^{4})\).
2. Multiply the coefficients together: \(-8 \times 2 \times -5\).
3. Apply the product rule for exponents to the variable parts: when multiplying like bases, add the exponents. So, add the exponents of \(t\): \$3 + 6 + 4$.
4. Combine the results from the coefficient multiplication and the variable exponent addition to write the simplified expression in the form \(a t^{b}\), where \(a\) is the product of coefficients and \(b\) is the sum of exponents.
5. Write the final simplified expression, ensuring the correct sign and exponent are included.

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, t³ × t⁶ × t⁴ equals t^(3+6+4) = t¹³. This property simplifies expressions with powers efficiently.

Handling Negative Signs in Multiplication

Multiplying negative numbers follows sign rules: the product of two negatives is positive, and the product of a positive and a negative is negative. Keep track of signs to determine the final sign of the product.