Verified Answer :

  • Multiply the coefficients: \(8 \times 3 = 24\).
  • Apply the product rule for exponents to the \(x\) variables: \(x^2 \times x = x^{2+1} = x^3\).
  • Keep the \(y\) term as is: \(y^3\).
  • Combine all parts to obtain the final simplified expression: \(\boxed{24x^3y^3}\).

Explanation

  1. Analysis of the Expression

The expression to be simplified is \(8x^2y^3 \times 3x\). We observe that it is a product of two monomials. To simplify this, we need to group the numerical coefficients and the variables separately and then apply the exponent rules. Specifically, the rule for multiplying powers with the same base states that \(x^a \times x^b = x^{a+b}\).

  1. Grouping and Simplifying Terms

First, we group the numerical coefficients:
\[8 \times 3 = 24.\]
Next, we group the variable \(x\) terms:
\[x^2 \times x = x^2 \times x^1 = x^{2+1} = x^3.\]
Finally, the variable \(y\) term, \(y^3\), has no corresponding \(y\) terms to multiply with, so it remains as \(y^3\).

  1. Final Calculation

By combining the results from the previous steps, we multiply the simplified coefficients, the simplified \(x\) terms, and the \(y\) terms together to get the final result:
\[24 \times x^3 \times y^3 = 24x^3y^3.\]
Thus, the simplified form of the expression is \(\boxed{24x^3y^3}\).

Examples

Simplifying expressions like \(8x^2y^3 \times 3x\) is a foundational skill in algebra, useful for solving complex equations and polynomial multiplication in engineering and physics. For example, if you are calculating the volume of an object where dimensions are given as algebraic expressions, simplifying these expressions helps you find the total volume or surface area efficiently, making design processes and material estimations much simpler to handle.

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