1. Convert the 5th root of x^3 into a rational expression.

2. Solve i^63

I don't know how to solve these so please help me!!!

x^(3/5)

i^2 = -1
i^3 = -i
i^4 = 1
i^5 = i
i^6 = -1
i^7 = -i
i^8 = 1
note pattern
i^64 = 1
and
i^63 = -i

Thank you so much. I understand it now!!

How did you get x^3/5?

fifth root of n = n^(1/5)

Thank you!!!

Sure! I'd be happy to help you with those questions.

1. To convert the 5th root of x^3 into a rational expression, we need to eliminate the radical. One way to do this is by using the property of radicals that states: the nth root of a product is equal to the product of the nth root of each factor.

To begin, we can rewrite the 5th root as a fractional exponent:

(x^3)^(1/5)

Now, applying the property of radicals mentioned above, we can rewrite it as:

(x^(3/5))

So the 5th root of x^3 can be expressed as the rational exponent x raised to the power of 3/5.

2. To solve i^63, we need to understand the properties of the imaginary unit "i", which is defined as the square root of -1.

The powers of "i" repeat in the following pattern:
i^1 = i
i^2 = -1
i^3 = -i
i^4 = 1

Since the powers of "i" repeat every four terms, we can divide the exponent by 4 to find the remainder. In this case, 63 divided by 4 is 15 with a remainder of 3.

So, i^63 can be written as i^3.
From the pattern above, we know that i^3 = -i.

Therefore, i^63 is equal to -i.

I hope that helps! Let me know if you have any further questions.