What operation do you use when applying the power of a product rule?

What operation do you use when applying the power of a product rule?

• A. Add all powers together
• B. Multiply each factor's power
• C. Subtract the powers of the factors
• D. Raise the entire product to the power

Answer: (B)

The power of a product rule states that when you raise a product (a multiplication of factors) to a power, you distribute that exponent to each factor in the product. This means that you take each base in the product and raise it individually to the given power. For example, if you have a product of two numbers, such as \( (a \cdot b)^n \), applying the power of a product rule involves raising both \( a \) and \( b \) to the power \( n \). The expression simplifies to \( a^n \cdot b^n \). This demonstrates that the correct operation is to independently raise each factor to the specified exponent, which is why the response indicating multiplication of each factor's power is accurate. This rule is a fundamental property of exponents that helps maintain the correct mathematical manipulation when dealing with expressions involving powers and products, ensuring the results remain consistent across algebraic operations.

The power of a product rule states that when you raise a product (a multiplication of factors) to a power, you distribute that exponent to each factor in the product. This means that you take each base in the product and raise it individually to the given power.

For example, if you have a product of two numbers, such as ( (a \cdot b)^n ), applying the power of a product rule involves raising both ( a ) and ( b ) to the power ( n ). The expression simplifies to ( a^n \cdot b^n ). This demonstrates that the correct operation is to independently raise each factor to the specified exponent, which is why the response indicating multiplication of each factor's power is accurate.

This rule is a fundamental property of exponents that helps maintain the correct mathematical manipulation when dealing with expressions involving powers and products, ensuring the results remain consistent across algebraic operations.