Exponents and Roots - SAT Math

First find all of the prime factors of

So

√75 can be broken down to √25 * √3. Which simplifies to 5√3.

Rewrite what is under the radical in terms of perfect squares:

Therefore, .

Multiply by the conjugate and the use the formula for the difference of two squares:

We know that 25 is a factor of 50. The square root of 25 is 5. That leaves which can not be simplified further.

1. We know that , which we can separate under the square root:

2. 144 can be taken out since it is a perfect square: . This leaves us with:

This cannot be simplified any further.

To simplify a square root, you have to factor the number and look for pairs. Whenever there is a pair of factors (for example two twos), you pull one to the outside.

Thus when you factor 96 you get

When simplifying square roots, begin by prime factoring the number in question. For , this is:

Now, for each pair of numbers, you can remove that number from the square root. Thus, you can say:

Another way to think of this is to rewrite as . This can be simplified in the same manner.

When simplifying square roots, begin by prime factoring the number in question. For , this is:

Now, for each pair of numbers, you can remove that number from the square root. Thus, you can say:

Another way to think of this is to rewrite as . This can be simplified in the same manner.

When simplifying square roots, begin by prime factoring the number in question. This is a bit harder for . Start by dividing out :

Now, is divisible by , so:

is a little bit harder, but it is also divisible by , so:

With some careful testing, you will see that

Thus, we can say:

Now, for each pair of numbers, you can remove that number from the square root. Thus, you can say:

Another way to think of this is to rewrite as . This can be simplified in the same manner.

We need to factor the number in the square root and find pairs of factors inorder to simplify a square root.

Since 83 is prime, it cannot be factored.

Thus the square root is already simplified.

If the triangle is a right triangle, then it follows the Pythagorean Theorem. Therefore:

--->

At this point, factor out the greatest perfect square from our radical:

Simplify the perfect square, then repeat the process if necessary.

Since is a prime number, we are finished!

There are two ways to solve this problem. If you happen to have it memorized that is the perfect square of , then gives a fast solution.

If you haven't memorized perfect squares that high, a fairly fast method can still be achieved by following the rule that any integer that ends in is divisible by , a perfect square.

Now, we can use this rule again:

Remember that we multiply numbers that are factored out of a radical.

The last step is fairly obvious, as there is only one choice:

A good method for simplifying square roots when you're not sure where to begin is to divide by , or , as one of these generally starts you on the right path. In this case, since our number ends in , let's divide by :

As it turns out, is a perfect square!

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:

This question may look daunting, especially if you start out by trying pick values or and solving. This plan of attack will work, but you're likely going to be dealing with some messy numbers. Instead we want to recall some of our exponent and root rules.

Let's look at this answer choice:

This double square root is the same as a fourth root. Think about it, you have a square times a square- which is the same thing as . Thus, so this choice can be eliminated.

Next, let's look at this answer choice:

For this choice, we need to recall our exponent rules. Remember, whenever we have a value raised to a fractional power, the denominator of that fraction is equal to the root number. In this case, . Thus,

Now, let's look at this answer choice:

A key rule to remember here is that order doesn't not matter when dealing with roots and powers. Thus, taking the root of a number and then cubing it will result in the same value as cubing a number and then taking the root .

This leaves us with:

If we tried to break this down a bit, we could take the third root of , which would leave us with:

. This will not equal .

We know that 25 is a factor of 50. The square root of 25 is 5. That leaves which can not be simplified further.

First find all of the prime factors of

So