To simplify a radical that has a negative sign under it we need to factor it first.

After factoring it recall that .

Therefore we get the following.

To simplify, we must FOIL:

Remembering that , we can simplify this further to

Rewrite the expression and eliminate the negative exponent.

Evaluate .

We can rewrite  as an exponential product using .

Rewrite the expression.

The answer is:  

i raised to each successive power has a different answer that repeats in a cycle of four.

Then, the cycle repeats with and and so on.

To solve any problem with i raised to a large number, the easiest way is to find the closest number that either is the exponent or is under the exponent that evenly divides by four and completes the cycle.

In this problem

The number 1680 divides evenly by 4. This means that 1680 completes the cycle and has the same answer as

i raised to each successive power has a different answer that repeats in a cycle of four.

Then, the cycle repeats with and and so on.

To solve any problem with i raised to a large number, the easiest way is to find the closest number that either is the exponent or is under the exponent that evenly divides by four and completes the cycle.

In this problem

The closest number under 358 that evenly divides by 4 is 356. 356 completes the cycle. 358 is two numbers into the start of a new cycle and has the same value as

i raised to each successive power has a different answer that repeats in a cycle of four.

Then, the cycle repeats with and and so on.

To solve any problem with i raised to a large number, the easiest way is to find the closest number that either is the exponent or is under the exponent that evenly divides by four and completes the cycle.

In this problem

The closest number under 273 that divides by 4 is 272. This means the next number 273 will be the start of a new cycle and have the same answer as

i raised to each successive power has a different answer that repeats in a cycle of four.

Then, the cycle repeats with and and so on.

To solve any problem with i raised to a large number, the easiest way is to find the closest number that either is the exponent or is under the exponent that evenly divides by four and completes the cycle.

In this problem

The closest number under the exponent that divides evenly by 4 is 700. This means that 703 is the third number in a new cycle and will have the same value as

Definition of

Thus,

Multiply the numerator and denominator by the conjugate of the denominator.

Multiply both the top and bottom by using the FOIL method.

Numerator:  

Recall that , which means that .

Denominator:

Divide the numerator with the denominator.

The answer is:  

Multiply the top and bottom by the conjugate of the denominator.

Multiply the numerator with the numerator.  Use the FOIL method.

Recall that , which indicates that .

Multiply the denominator with the denominator.

Divide the numerator with the denominator.

The answer is: