Final answer:
The expression 4√5 can be written with a rational exponent by converting the radical into a fractional exponent, resulting in 4(51/2). The coefficient 4 remains unchanged and is multiplied by the radical expressed as 5 to the power of 1/2.
Step-by-step explanation:
The student has asked how to write 4√5 using rational exponents. The notation 4√5 can be confusing because it combines a coefficient (4) and a radical. To express this using a rational exponent, we focus on the radical part, which is the square root of 5. The square root of a number can be written with a fractional exponent of 1/2. Therefore, √5 is equivalent to 51/2.
However, the original expression has the coefficient of 4, which needs to be multiplied by the radical. Thus, 4√5 would be written as 4(51/2). This coefficient (4) stays as it is because it is not affected by the radical. Keep in mind that the coefficient should not be confused with an exponent; it is merely a number by which the radical is multiplied. Therefore, the correct answer with a rational exponent is option b. 4(51/5).
Please note that none of the other options provided are correct because they do not properly represent the multiplication of 4 by the square root of 5, nor do they apply the rational exponent correctly to the radical part of the expression.