Complex Numbers - PSAT Math

Rewrite in fraction form, and multiply both numerator and denominator by :

Multiply both numerator and denominator by :

Multiply the numerator and the denominator by the complex conjugate of the denominator, which is :

Multiply the numerator and the denominator by the complex conjugate of the denominator, which is :

Multiply the numerator and the denominator by the complex conjugate of the denominator, which is :

Multiply the numerator and the denominator by the complex conjugate of the denominator, which is . The above becomes:

Multiply the numerator and the denominator by the complex conjugate of the denominator, which is . The above becomes:

, so 56 is a multiple of 4. raised to the power of any multiple of 4 is equal to 1, so

.

This is the product of a complex number and its complex conjugate. They can be multiplied using the pattern

with

By the power of a product property,

The first step to solving this problem is distributing the exponent:

Next, we need simplify the complex portion.

Thus, our final answer is .

Use the FOIL method that states to multiply the Firsts, Outter, Inner, Lasts. Also remember that :

raised to the power of any multiple of 4 is equal to 1, so the above expresion is equal to

This is not among the given choices.

You are being asked to evaluate

You can use the cube of a binomial pattern with :

can be calculated by squaring , then squaring the result, using the square of a binomial pattern as follows:

The product of any complex number and its complex conjugate is the real number , so all that is needed here is to evaluate the expression:

First, square using the square of a binomial pattern as follows:

Raising this number to the fourth power yields the correct response:

To raise a negative number to an odd power, take the absolute value of the base to that power and give its opposite:

To raise to a power, divide the power by 4 and raise to the remainder. Since

,

Therefore,