If , then by the zero product principle, one or both of  and  is equal to 0. Since  is positive, . Also, since  is positive,  is positive, and . Thus, .

When breaking down a radical, we first want to find the largest perfect square that might be a factor for the number under the radical. 

We start with 4, 9, 16, 25 etc. until we find which one is a factor. 

In this case, 4 is a factor of 24. 

We can now break down the radical to become: 

The square root of 4 becomes 2 and the square root of 6 will not break down any further, this leads us to the answer below: 

When breaking down a square root, we must first find the largest perfect square factor that goes into the number under the radical; starting with 4, 9, 16, 25, 36 etc. 

In this case, 36 will go into 72, 2 times. 

Which reduces the radical to the below: 

We can then simplify square root 36 to become 6 and we get: 

When we multiply with a radical, only the numbers outside the radical are multiplied.

We must first simplify each radical by finding the largest perfect square that will go into each number starting with 4, 9, 16, 25 etc. 

For 12, the perfect square of 4 is a factor. 

For 27, the perfect square of 9 is a factor. 

Simplifying both square roots gives us: 

Simplifying the above becomes: 

Both parts of this expression contain a radical 3 which means they are like terms which can be added. 

When we add the 2 and the 3 we get the below: 

Solve each exponent:

Find the sum:

Find the square root:

The rolls that yield a product greater than or equal to 20 are:

These are 8 out of 36 rolls, so the probability of getting one of them is 

Consider the different ways in which you could roll an 8 or a 7.

You could roll 8 by the following combinations:

2-6, 3-5, 4-4, 5-3, 6-3

so the odds of rolling 8 are .

You could roll 7 by the following combinations:

1-6, 2-5, 3-4, 4-3, 5-2, 6-1

so the odds of rolling 7 are .

The answer, therefore, is that Column B is greater.

With the replacement of the tens with the queens, there are still 52 cards in the deck, but now, there are four jacks, eight queens, and four kings - 16 face cards. The probability that a random card is a face card is 

The removal of the two black aces leaves a deck of 50 cards, with all 12 face cards remaining. The probability that a randomly drawn card is a face card is therefore

Since  , the probability is less than  .

Since each die is fair, the roll of each die is an independent event; also, the second roll of the blue die is independent of the first roll. Therefore, the probabilities of each outcome are the same for Sandy rolling two dice simultaneously as for Tommy rolling one twice. (a) and (b) are equal.