Final answer:
The value of "?" that makes the equation (2a^4b^2)^? (3ab^4) = 24a^13b^10 true is 3, which is confirmed by the coefficients and the exponents of the variables on both sides of the equation.
Step-by-step explanation:
To determine the value of "?" that makes the equation (2a^4b^2)^? (3ab^4) = 24a^13b^10 true, we need to apply the rules of exponents to equate the bases and their respective powers. First, let's express the equation with the unknown exponent applied to both the coefficient and the variables:
- (2^? * a^(4?) * b^(2?)) * (3 * a * b^4)
We then multiply the coefficients and add the exponents of the like bases to set them equal to the corresponding parts of the right side of the equation:
- (2^? * 3) * a^(4? + 1) * b^(2? + 4) = 24 * a^13 * b^10
Now we can see that the coefficient must be 24 when 2^? is multiplied by 3. Accounting for this, we recognize that 2^3 * 3 = 8 * 3 = 24, so ? must be 3. Moreover, for the variable a, the exponents on both sides must be equal (4? + 1 = 13), which simplifies to 4 * 3 + 1 = 13, confirming that ? is 3. For the variable b, the exponents must also match (2? + 4 = 10), which simplifies to 2 * 3 + 4 = 10, again confirming that the value of ? is 3.
Therefore, using the rules of exponents, we can conclude that Option 1: ? = 3 is the correct answer, as it is the only value that satisfies the equation for both variables a and b.