# SAT Math : How to multiply complex numbers

## Example Questions

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### Example Question #40 : Complex Numbers

Raise  to the fourth power.

None of these

Explanation:

By the Power of a Power Rule, the fourth power of any number is equal to the square of the square of that number:

Therefore, one way to raise  to the fourth power is to square it, then to square the result.

Using the binomial square pattern to square :

Applying the Power of a Product Property:

Since  by definition:

Square this using the same steps:

### Example Question #51 : Squaring / Square Roots / Radicals

Raise  to the fourth power.

None of these

Explanation:

The easiest way to find  is to note that

.

Therefore, we can find the fourth power of  by squaring , then squaring the result.

Using the binomial square pattern to square :

Applying the Power of a Product Property:

Since  by definition:

Square this using the same steps:

Therefore,

### Example Question #41 : Complex Numbers

Raise  to the third power.

None of these

Explanation:

To raise any expression  to the third power, use the pattern

Setting :

Taking advantage of the Power of a Product Rule:

Since  and :

Collecting real and imaginary terms:

### Example Question #51 : Squaring / Square Roots / Radicals

Evaluate:

The expression is undefined

Explanation:

is defined to be equal to  for any real or imaginary  and for any real ; therefore,

To evaluate a positive power of , divide the power by 4 and note the remainder:

Therefore,

Substituting,

Rationalizing the denominator by multiplying both numerator and denominator by :

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