Grade what is the units digit of the simplest form of ? Answer: Solution

Answer: 12
Solution:
81,000 = 2³ x 3⁴ x 5³
In 2³ 2⁰, 2² are the perfect square divisors
In 3⁴ 3⁰, 3², 3⁴ are the perfect square divisor
In 5³ 5⁰, 5² are the perfect square divisors
By counting principle, there are 2 x 3 x 2 = 12 perfect square divisors

7. 
   X, Y, and Z are prime numbers and X < Y < Z. What is the value of Y in
   

3X + 2Y + 5 Z = 143?
Answer: 11
Solution:
The sum is an odd number. The possible combinations to give an odd answer is
Even + even + odd = odd
Odd + odd + odd = odd
For the first case, even + even + odd = odd, the only even number is 2 and when multiplied to 3 or 5 will give an even addends. Since x < y < z, x = 2. If x = 2, 2y + 5z = 137 or 5z = 137 – 2y, which implies that y should be a prime number that ends with 1. Thus could be y= 11, 31, . . . and z= 23, 15 . . . respectively. The only possible value of y and z that are prime are y=11, z= 23.
For the second case odd + odd + odd = odd
This is impossible to get since the 2y will give us an even addend.
Therefore, the only value for Y is 11.

8. 
   If x = , determine the value of .
   

Ans.
Solution:
x = = , x = 3 +
x² = 14 + 6 , x – 3 =
= = = =

9. 
   Simplify (x⁶⁴ + 1) (x³² + 1) (x¹⁶ + 1). . . (x⁴ + 1)(x² + 1)(x + 1) and find its value when x = 2. Express your answer in exponential form.
   

Ans.
Solution:

(x⁶⁴ + 1) (x³² + 1) (x¹⁶ + 1). . . (x⁴ + 1)(x² + 1)(x + 1) =

If x = 2, then = =

10. 
    If x is a real number so that 2x² – x - 2 = 0, find the value of .