Again here, if no perfect square is easily recognized try dividing by , , or .

Note that the  we obtained by simplifying  is multiplied, not added, to the  already outside the radical.

To solve, simply find a perfect square factor and pull it out of the square root.

Recall the factors of 48 include (16, 3). Also recall that 16 is a perfect square since 4*4=16.

Thus,

The trick to these problems is to simplify the radical by using the following rule:   and  Here, we need to find a common factor for the radical. This turns out to be five because  Remember, we want to include factors that are perfect squares, which are what nine and four are. Therefore, we can rewrite the equation as: 

When simplifying the square root of a number that may not have a whole number root, it's helpful to approach the problem by finding common factors of the number inside the radicand. In this case, the number is 24,300.

What are the factors of 24,300?

24,300 can be factored into:

When there are factors that appear twice, they may be pulled out of the radicand. For instance, 100 is a multiple of 24,300. When 100 is further factored, it is  (or 10x10). However, 100 wouldn't be pulled out of the radicand, but the square root of 100 because the square root of 24,300 is being taken. The 100 is part of the24,300. This means that the problem would be rewritten as:  

But 243 can also be factored: 

Following the same principle as for the 100, the problem would become
 because there is only one factor of 3 left in the radicand. If there were another, the radicand would be lost and it would be 9*10*3. 
9 and 10 may be multiplied together, yielding the final simplified answer of 

To solve the equation , we can first factor the numbers under the square roots.

When a factor appears twice, we can take it out of the square root.

Now the numbers can be added directly because the expressions under the square roots match.

In order to solve for x, we must first take the square root of both sides.

(x - 3)² = 64

(x - 3) = +/- 8

The square root of 64 is either 8 or -8. We then solve the equation with both possible values of the square root of 64.

x - 3 = 8 (Add 3 to both sides.)

x = 11

x - 3 = -8 (Add 3 to both sides.)

x = -5

x = 11 or -5. -5 is the only possible value of x that is an answer choice. 

When using a cube root. 3³ becomes 3 * 3 * 3.

3 * 3 = 9

9 * 3 = 27

A rational number is any number than can be expressed as a fraction or a quotient of two integers. The square of π is still an irrational number. (√(2))² = 2, (1/3)² = 1/9, (7/17)² = 49/289, (1/2)² = 1/4, all can be writen in fractional form, and are thus, rational numbers. 

The square of –2 is 4, of –3 is 9, and of 4 is 16.

The sum of these squares is 29.

The sum of (–2) + (–3) + (4) is –1.

29 – (–1) = 30

 falls between  and  

Therefore,  is between  and