Square Roots and Operations - ACT Math

Find the Sum:

Simplify the radicals:

The first step when adding square roots is to simplify each term as much as possible. Since the first term has a square within the square root, we can reduce to . Now each term has only prime numbers within the sqaure roots, so nothing can be further simplified and our new expression is .

Only terms that have the same expression within the sqaure root can be combined. In this question, these are the first and third terms. When combining terms, the expression within the square root stays the say, while the terms out front are added. Therefore we get . Since the second term of the original equation cannot be combined with any other term, we get the final answer of .

Begin by factoring out each of the radicals:

For the first two radicals, you can factor out a or :

The other root values cannot be simply broken down. Now, combine the factors with :

This is your simplest form.

Begin by getting your terms onto the left side of the equation and your numeric values onto the right side of the equation:

Next, you can combine your radicals. You do this merely by subtracting their respective coefficients:

Now, square both sides:

Solve by dividing both sides by :

Begin simplifying by breaking apart the square roots in question. Thus, you know:

Now, with square roots, you can combine factors just as if a given root were a variable. So, just as , so too does .

Begin simplifying by breaking apart the square roots in question. Thus, you know:

Now, with square roots, you can combine factors just as if a given root were a variable. So, just as , so too does .

Upon dividing by , the in the numerator cancels out, and the only number remaining is :

Find the greatest factor that is a perfect square for 112 and 63. For 112, the factors are 16 and 7, thus . For 63, the factors are 9 and 7, thus . Simplifying these terms will give . Cancel out the results in .

Although it may seem as though we cannot do anything to this expression due to our numerator and denominator having different indices, there is in fact some real simplifying to be done here.

To begin, our numerator can be evaluated, because 343 is in fact a perfect cube:

This fact helps us out with our denominator as well. Our original equation becomes the following:

Then, we can pull out two of the sevens on the bottom and cancel them like so:

We may seem to be done, but if you look, you will not see our solution among the answer choices. That is because we need to rationalize the denominator. We should never have a square root in our denominator. To remedy this, we do a fairly simple move that won't change the value of our fraction, but just the form it is in:

Now we'll have no issue finding our answer choice!

Division of square roots is easy, since you can combine the roots and treat it like any other fraction. Thus, you can convert our fraction as follows:

Next, you begin to reduce the fraction:

This reduces to:

Now, break this apart again into:

, which is

Division of square roots is easy, since you can combine the roots and treat it like any other fraction. Thus, you can convert our fraction as follows:

Next, you begin to reduce the fraction:

This reduces to:

Now, break this apart again into:

, which is

Finish by rationalizing the denominator:

It might be a little difficult taking a fourth root of 100 to isolate x by itself; it might be easier to select an integer and take that number to the fourth power. For example 34 = 81 and 44 = 256. Since 34 is less than 100 and 44 is greater than 100, x would lie between 3 and 4.

To find a ratio like this, you need to divide by . Recall that when you have the square root of a fraction, you can "distribute" the square root to the numerator and the denominator. This lets you rewrite as:

Next, you can write the ratio of the two variables as:

Now, when you divide by a fraction, you can rewrite it as the multiplication by the reciprocal. This gives you:

Simplifying, you get:

You should rationalize the denominator:

This is the same as:

The ratio of two numbers is merely the division of the two values. Therefore, for the information given, we know that the ratio of

to

can be rewritten:

Now, we know that the square root in the denominator can be "distributed" to the numerator and denominator of that fraction:

Thus, we have:

To divide fractions, you multiply by the reciprocal:

Now, since there is one in , you can rewrite the numerator:

This gives you:

Rationalize the denominator by multiplying both numerator and denominator by :

Let's be careful how we write the numerator so as to make explicit the shared factors:

Now, reduce:

This is the same as

Simplify the radicals, then multiply:

To simplify this expression, simply multiply the radicands and reduce to simplest form.

When multiplying square roots, the easiest thing to do is first to factor each root. Thus:

Now, when you combine the multiplied roots, it will be easier to come to your final solution. Remember that multiplying roots is very easy! Just multiply together everything "under" the roots:

Finally this can be simplified as:

When multiplying square roots, the easiest thing to do is first to factor each root. Thus:

Now, when you combine the multiplied roots, it will be easier to come to your final solution. Just multiply together everything "under" the roots:

Finally this can be simplified as:

Don't try to do too much at first for this problem. Multiply your radicals and your coefficients, then worry about any additional simplification.

Now simplify the radical.