Which exponent rule(s) would be used to simplify this expression? 2x^5 ⋅ 12x^−4

Question

The power to power rule since you have two different exponents on the x, so the new exponent would be -20. Also, you would add 2 and 12 using the product rule for exponents.
The power to power rule since you have two different exponents on the x, so the new exponent would be -20. Also, you would add 2 and 12 using the product rule for exponents.

The negative exponent rule to get 5 - (-4) = 9 for the power of x and then multiply 2 x 12 = 24 for the coefficient.
The negative exponent rule to get 5 - (-4) = 9 for the power of x and then multiply 2 x 12 = 24 for the coefficient.

The quotient rule to subtract 2 - 12 = -10. You would also subtract the exponents to get 5 - 4 = 1 for the power of x.
The quotient rule to subtract 2 - 12 = -10. You would also subtract the exponents to get 5 - 4 = 1 for the power of x.

The product exponent rule to add 5 + -4 to get an exponent of 1 on the x. You would also multiply 2 x 12 since they are coefficient

Answer 1

To simplify the expression \( 2x^5 \cdot 12x^{-4} \), the correct rules to apply are:

1. Product of Coefficients: Multiply the numerical coefficients (2 and 12).
2. Product of Powers Rule: When multiplying like bases, you add the exponents.

Here’s a step-by-step breakdown:

1. Multiply the coefficients: \[ 2 \cdot 12 = 24 \]

2. Apply the product of powers rule for \(x\): \[ x^5 \cdot x^{-4} = x^{5 + (-4)} = x^{5 - 4} = x^1 = x \]

3. Combine the results: \[ 2x^5 \cdot 12x^{-4} = 24x^1 = 24x \]

Therefore, the appropriate rules used here are the product of coefficients and the product of powers rule.

So, the last option ("The product exponent rule to add 5 + -4 to get an exponent of 1 on the x. You would also multiply 2 x 12 since they are coefficients.") is the correct choice.