- Identify the terms with the same base in the expression \(-2 x^3 y^4 x^2 y^3\).
- Group and simplify using the product of powers property \(a^m \cdot a^n = a^{m+n}\).
- Calculate the new exponents: \(x^{3+2} = x^5\) and \(y^{4+3} = y^7\).
- Final simplified expression is \(-2 x^5 y^7\), which matches option b.
- Final answer: \(\boxed{\text{b}}\)
Explanation
- Problem Analysis
The given expression to simplify is \(-2 x^3 y^4 x^2 y^3\). To simplify this, we rely on the commutative and associative properties of multiplication, which allow us to rearrange the terms and group variables with the same base. We also utilize the product of powers property, which states that for any real number \(a\) and integers \(m\) and \(n\), \(a^m \cdot a^n = a^{m+n}\).
- Simplification Process
First, rearrange the expression by grouping the numerical coefficient and the variables \(x\) and \(y\) together:
\[-2 \cdot (x^3 \cdot x^2) \cdot (y^4 \cdot y^3)\]
Next, apply the product of powers property \(a^m \cdot a^n = a^{m+n}\) to combine the exponents of each variable:
- For \(x\): \(x^3 \cdot x^2 = x^{3+2} = x^5\)
- For \(y\): \(y^4 \cdot y^3 = y^{4+3} = y^7\)
Substituting these back into the expression, we obtain:
\[-2 \cdot x^5 \cdot y^7 = -2 x^5 y^7\]
- Conclusion and Final Answer
Comparing the simplified expression \(-2 x^5 y^7\) with the given options:
A\(-2 x^6 y^{12}\)
B\(-2 x^5 y^7\)
C\(-2(x y)^{12}\)
D\(-2(x y)^{72}\)
The calculated result matches option b. Therefore, the correct simplified form is \(-2 x^5 y^7\), which corresponds to option b. The final answer is \(\boxed{\text{b}}\).
Examples
The product of powers rule is essential in fields like physics and engineering, where complex formulas involve multiplying quantities expressed as powers of the same base. For example, when calculating the area or volume of shapes where dimensions are defined by variable expressions (like \(l = 2x^3\) and \(w = x^2\)), you simplify the expression to determine the total area as \(2x^5\). Mastering this rule helps in simplifying algebraic expressions quickly and accurately, forming the foundation for more advanced calculus and algebra.