- Use the commutative and associative properties to rearrange the expression: \((7 \cdot 3)(x^2 \cdot x^5)(y^3 \cdot y^8)\).
- Multiply the coefficients: \(7 \cdot 3 = 21\).
- Apply the product rule for exponents \(x^a \cdot x^b = x^{a+b}\) to the variables: \(x^2 \cdot x^5 = x^7\) and \(y^3 \cdot y^8 = y^{11}\).
- Combine the results to get the product \(21x^7y^{11}\), which corresponds to option \(\boxed{c}\).
Explanation
- Decomposing the Expression
To find the product of the monomials \((7x^2 y^3)\) and \((3x^5 y^8)\), we recognize that the expression is a multiplication of four distinct parts: the constant coefficients, the powers of \(x\), and the powers of \(y\). According to the associative and commutative properties of multiplication, we can rearrange the factors as follows:
\[(7x^2 y^3)(3x^5 y^8) = (7 \cdot 3) \cdot (x^2 \cdot x^5) \cdot (y^3 \cdot y^8)\]
This allows us to address the coefficients and each variable separately.
- Multiplying the Coefficients
Next, we calculate the product of the constant coefficients:
\[7 \cdot 3 = 21\]
This gives us the coefficient for our final expression.
- Applying Exponent Rules
For the variables, we apply the exponent rule \(x^a \cdot x^b = x^{a+b}\).
For the base \(x\):
\[x^2 \cdot x^5 = x^{2+5} = x^7\]
For the base \(y\):
\[y^3 \cdot y^8 = y^{3+8} = y^{11}\]
Combining these results, we get the product of the variable parts.
- Conclusion and Selection
By multiplying the coefficient, the power of \(x\), and the power of \(y\) together, we form the complete product:
\[21 \cdot x^7 \cdot y^{11} = 21x^7y^{11}\]
Comparing this result to the given options:
A\(21 x^{10} y^{24}\)
B\(10 x^7 y^{11}\)
C\(21 x^7 y^{11}\)
D\(10 x^{10} y^{24}\)
The calculated product corresponds to option c. Therefore, the final answer is \(\boxed{c}\).
Examples
In physics, when calculating physical quantities like momentum or energy, you often multiply expressions containing variables with exponents. For instance, if you are determining the area of a rectangle where the length is \(3x^5y^8\) and the width is \(7x^2y^3\), finding the total area involves multiplying these two monomials, much like in our problem, to determine the final expression for the area in terms of the variables \(x\) and \(y\). Understanding these exponent rules is essential for simplifying algebraic expressions in various scientific and engineering fields.