# ACT Math : Square Roots and Operations

## Example Questions

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### Example Question #1 : How To Multiply Square Roots

Find the product:

Explanation:

### Example Question #2 : How To Multiply Square Roots

Simplify the following completely:

Explanation:

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

### Example Question #1 : How To Multiply Square Roots

Simplify:

Explanation:

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:

### Example Question #4 : How To Multiply Square Roots

Simplify the following:

Explanation:

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:

### Example Question #5 : How To Multiply Square Roots

State the product:

Explanation:

### Example Question #6 : How To Multiply Square Roots

Find the product:

Explanation:

### Example Question #1 : How To Find A Ratio Of Square Roots

x= 100

If x is placed on a number line, what two integers is it between?

2 and 3

5 and 6

4 and 5

3 and 4

Cannot be determined

3 and 4

Explanation:

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 3= 81 and 4= 256. Since 34 is less than 100 and 44 is greater than 100, x would lie between 3 and 4.

### Example Question #2 : How To Find A Ratio Of Square Roots

What is the ratio of  to ?

Explanation:

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

### Example Question #3 : How To Find A Ratio Of Square Roots

and

What is the ratio of  to ?

Explanation:

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:

Find the sum: