Simplify each exponential expression in Exercises 23–64. (−5x^4 y)(−6x^7 y^11)

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Identify the expression to simplify: $(-5x^4 y)(-6x^7 y^{11})$.

Apply the associative property of multiplication to group the coefficients and like bases: $(-5) \cdot (-6)$, $x^4 \cdot x^7$, and $y \cdot y^{11}$.

Multiply the coefficients: $(-5) \cdot (-6)$.

Use the product of powers property for the $x$ terms: $x^4 \cdot x^7 = x^{4+7}$.

Use the product of powers property for the $y$ terms: $y \cdot y^{11} = y^{1+11}$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Exponential Properties

Exponential properties govern how to manipulate expressions involving exponents. Key rules include the product of powers rule, which states that when multiplying like bases, you add the exponents (a^m * a^n = a^(m+n)). Understanding these rules is essential for simplifying expressions that contain variables raised to powers.

Multiplication of Polynomials

Multiplying polynomials involves distributing each term in one polynomial to every term in the other. In the expression (−5x^4 y)(−6x^7 y^11), you multiply the coefficients and then apply the properties of exponents to the variable parts. This process is crucial for simplifying expressions that contain multiple variables and exponents.

Combining Like Terms

Combining like terms is the process of simplifying expressions by adding or subtracting terms that have the same variable raised to the same power. In the context of the given expression, after multiplication, you will need to identify and combine any like terms that may arise, ensuring the final expression is in its simplest form.

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