In order for the original statement to be true, the  and  terms must be either both odd or both even. Looking at each of the statements individually,

I. States that  is odd, but only odd values multiplied by 3 are odd. If  was an even number, the result would be even. But can be either odd or even, depending on what equals. Thus this statement COULD be true but does not HAVE to be true.

II. States that  is an integer, and since only even numbers are cleanly divided by 2 (odd values result in a fraction) this ensures that  is even. However, can also be odd, so this is a statement that COULD be true but does not HAVE to be true.

III. For exponents, only the base value determines whether it is even or odd - it does not indicate the value of y at all. Only even numbers raised to any power are even, thus, this ensures that  is even. But can be odd as well, so this statement COULD be true but does not HAVE to be true.

An example of two integers that will work violate conditions II and III is and .

, and even number.

is not an integer.

is not even.

Furthermore, any combination of 2 even integers will make the original statement true, and violate the Statement I.

First list all the integers between 20 and 80 that are squares of another integer:

5² = 25

6² = 36

7² = 49

8² = 64

In total, there are 61 integers from 20 to 80, inclusive. 61 – 4 = 57

Notice how is the greatest value. This often means that is negative as when  is odd and when  is even.

Let us examine the first choice,

This can only be true of a negative value that lies between zero and one.

To solve for  you have to get all the 's on one side and all the numbers on the other side.

Combine like terms, and the result is 

.

Divide by  on both sides to get 

.

Finally take the square root of both sides to get .

This question is testing one's ability to analyze a function algebraically and recognize different but equivalent forms. Identifying properties of functions through analyzing equivalent forms is critical to this concept. Such properties that can be found through analyzing the different forms of a function include finding roots (zeros), extreme values, symmetry, and intercepts.

Knowing the standard and the concept for which it relates to, we can now do the step-by-step process to solve the problem in question.

Step 1: Recognize the general form of the function.

This is known as the difference of squares.

Step 2: Identify what is known.

Step 3: Substitute the known values into the difference of perfect squares found in step 1.

Step 4: Answer the question.

To find the positive solution set each binomial equal to zero and solve for x.

Therefore the positive solution is four.

Given the two equations, substitute the numerical value of  into the second equation to solve for . 

Substituting the numerical value for  into the equation with  is as follows.

From here, distribute the three.

Now square both side of the equation.

Remember to square both terms within the parentheses. Also, recall that squaring a square root sign cancels them out.

To solve for  divide both sides by six.

22% = 22/100

Divide everything by 2:

22/100 = 11/50

11 is a prime number, so this is as reduced as this fraction can get.

The problem states that decreasing y by ten percent gives us the same thing as taking fifteen percent of x. We need to find an expression for decreasing y by ten percent, and an expression for fifteen percent of x, and then set these two things equal.

If we were to decrease y by ten percent, we would be left with ninety percent of y (because the percentages must add to one hundred percent). We could write ninety percent of y as 0.90y = (90/100)y = (9/10)y. Remember, when converting from a percent to a decimal, we need to move the decimal two places to the left.

Similarly, we can write 15% of x as 0.15x = (15/100)x = (3/20)x.

Now, we set these two expressions equal to one another.

(9/10)y = (3/20)x

Multiply both sides by 20 to eliminate fractions.

18y = 3x

The question asks us to find the ratio of x to y, which is equal to x/y. Thus, we must rearrange the equation above until we have x/y by itself on one side.

18y = 3x

Divide both sides by 3.

6y = x

Divide both sides by y.

6 = x/y

Thus, the ratio of x to y is 6.

The answer is 6.

First convert the percentage to a decimal:

7.5% = .075

Then turn this into a fraction:

.075 = 75/1000

Simplify by dividing the numerator and denominator by 25:

75/1000 = 3/40

25% of 64 is 16 (you can find this with a calculator by 0.25 * 64). Divide 16 by 0.05 (or 1/20) to get the value of the number 16 is 5% of. (Or mental math of 16 * 20)