# SAT II Math I : Irrational Numbers

## Example Questions

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### Example Question #2 : Irrational Numbers

Which of the following is not an irrational number?

Explanation:

A root of an integer is one of two things, an integer or an irrational number. By testing all five on a calculator, only  comes up an exact integer - 5. This is the correct choice.

### Example Question #1 : Irrational Numbers

Simplify by rationalizing the denominator:

Explanation:

Multiply the numerator and the denominator by the conjugate of the denominator, which is . Then take advantage of the distributive properties and the difference of squares pattern:

Explanation:

### Example Question #8 : Number Theory

Multiply:

Explanation:

Use the FOIL technique:

### Example Question #1 : Imaginary Numbers

Evaluate:

Explanation:

We can set  in the cube of a binomial pattern:

### Example Question #1 : Irrational Numbers

Evaluate

You cannot divide by complex numbers

Explanation:

To divide by a complex number, we must transform the expression by multiplying it by the complex conjugate of the denominator over itself. In the problem,  is our denominator, so we will multiply the expression by  to obtain:

.

We can then combine like terms and rewrite all  terms as . Therefore, the expression becomes:

### Example Question #1 : Imaginary Numbers

Simplify the following product:

Explanation:

Multiply these complex numbers out in the typical way:

and recall that  by definition. Then, grouping like terms we get

### Example Question #2 : Imaginary Numbers

Identify the real part of

none of the above.

Explanation:

A complex number in its standard form is of the form: , where  stands for the real part and  stands for the imaginary part. The symbol  stands for .

The real part in this problem is 1.

### Example Question #3 : Imaginary Numbers

Simplify:

Explanation:

To add complex numbers, find the sum of the real terms, then find the sum of the imaginary terms.

Simplify: