show all the steps that you use to solve this problem in the space provided. Simplify 46√30√ by rationalizing the denominator. Show your work
To simplify the expression 46√30, we first break down 30 into its prime factors:
30 = 2 * 3 * 5
Next, we simplify the square root:
√30 = √(2 * 3 * 5) = √2 * √3 * √5 = √2√3√5 = √(2 * 3 * 5) = √(30)
Therefore, 46√30 = 46√(2 * 3 * 5) = 46√(30)
To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator:
46√30 / √30 = 46√30 / √30 * (√30 / √30) = 46√30√30 / 30 = 46√30√30 / 30
This simplifies to 46√30√30 / 30 = 46 * 30 / 30 = 46
Therefore, the expression simplifies to 46.