Problem: | 1 | 2 | 3 | 4 | Total |

Score: |

- (25 points) Write a careful proof of the irrationality of
the square root of 7.
- (25 points)
- Write 1.234523452345..(2345 repeats) as a rational number
in the form
`p/q`,`p,q`integers. - Prove or give a counterexample: A real number has a repeating
*binary*``decimal'' expansion if and only if it is a rational.

- Write 1.234523452345..(2345 repeats) as a rational number
in the form
- (25 points)
- Use the geometric interpretation of complex multiplication
to locate
`i`in the complex plane.^{3} - Use the geometric interpretation of complex
multiplication to locate the three cube roots of
`8i`.

- Use the geometric interpretation of complex multiplication
to locate
- (25 points)
- Calculate the inverse of the matrix
[4 3] [2 2]

- Use this inverse to solve the system of equations:
4x + 3y = 1 2x + 2y = 2

- Calculate the inverse of the matrix