Verified Answer :

  • Multiply the numerical coefficients: \((-2) \times (-8) = 16\).
  • Multiply the variable parts using the law of exponents: \(s \times s^3 = s^4\).
  • Combine both parts to arrive at the final simplified expression \(\boxed{16s^4}\).

Explanation

  1. Understanding the Expression

We need to simplify the expression \((-2s)(-8s^3)\). This involves multiplying two monomials. The expression consists of coefficients \(-2\) and \(-8\) and variable terms \(s^1\) and \(s^3\). According to the rules of algebra, we can use the associative and commutative properties of multiplication to group the coefficients and the variable parts.

  1. Multiplying the Coefficients

To simplify the expression, we first multiply the numerical coefficients:
\[(-2) \times (-8) = 16\]
Since both coefficients are negative, their product is positive.

  1. Multiplying the Variables

Next, we multiply the variable parts using the laws of exponents. The law states that for any base \(x\) and exponents \(m\) and \(n\), \(x^m \times x^n = x^{m+n}\). Applying this to our variables:
\[s^1 \times s^3 = s^{1+3} = s^4\]

  1. Combining the Results

Finally, we combine the product of the coefficients and the product of the variables to obtain the simplified expression:
\[16 \times s^4 = 16s^4\]
Thus, the simplified form is \(\boxed{16s^4}\).

Examples

In physics and engineering, simplified expressions are vital for calculations involving energy or area. For instance, if you are calculating the area of a rectangle where the width is represented as \(2s\) and the length is represented as \(8s^3\), the total area would be \((2s)(8s^3) = 16s^4\). This simplification makes it much easier to determine how the area changes as the dimension \(s\) scales.