### All ACT Math Resources

## Example Questions

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

x^{4 }= 100

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

**Possible Answers:**

Cannot be determined

4 and 5

2 and 3

5 and 6

3 and 4

**Correct answer:**

3 and 4

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^{4 }= 81 and 4^{4 }= 256. Since 3^{4} is less than 100 and 4^{4} is greater than 100, x would lie between 3 and 4.

### Example Question #7 : Square Roots And Operations

What is the ratio of to ?

**Possible Answers:**

**Correct answer:**

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 #8 : Square Roots And Operations

and

What is the ratio of to ?

**Possible Answers:**

**Correct answer:**

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:

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