When we have two identical numbers, each raised to an exponent, and multiplied together, we add the exponents together:

x^ax^b = x^a+b

This means that (0.5)³(0.5)³ = (0.5)³⁺³ = (0.5)⁶

Because 0.5 is between 0 and 1, we know that when it is multipled by itself, it decreases in value. Example: 0.5 * 0.5 = 0.25. 0.5 * 0.5 * 0.5 = 0.125. Etc.

Thus, (0.5)⁶ > (0.5)⁷

Answer: The relationship cannot be determined from the information provided.

Explanation: The best thing to do here is to notice that quantity A is composed of two complex terms with odd exponents. Odd powers result in negative results when their base is negative. Thus quantity A will be negative when either x or y (but not both) is negative. Otherwise, quantity A will be positive. Quantity B, however, has two even exponents, meaning that it will always be positive. Thus, sometimes Quantity A will be greater and sometimes Quantity B will be greater. Thus the answer is that the relationship cannot be determined.

Let's do each of these separately:

x³ * 2x⁴ * 5y = 2 * 5 * x³ * x⁴ * y = 10 * x⁷ * y = 10x⁷y

4y² + 3y² = 7y²

Now, rewrite what we have so far:

(10x⁷y + 7y²)/y

There are several options for reducing this.  Remember that when we divide, we can "distribute" the denominator through to each member.  That means we can rewrite this as:

(10x⁷y)/y + (7y²)/y

Subtract the y exponents values in each term to get:

10x⁷ + 7y

First let's look at Quantity B:

(x/3)³ = x³/27. Now both columns have an x³ so we can cancel it from both terms. Therefore we're now comparing 1/3 in Quantity A to 1/27 in Quantity B.  1/3 is the larger fraction so Quantity A is greater.

However, if , then the two quantities would both equal 0.  Thus, since the two quantities can have different relationships based on the value of , we cannot determine the relationship from the information given.

The two quantites are equal. To take the exponent of an exponent, the two exponents should be multiplied.

(2³ )² or 2^3*2 = 64   

(2² )³ or 2^2*3 = 64   

Both quantities equal 64, so the two quantities are equal.

To compare these expressions more easily, we'll change the first expression to have  in front. We'll do this by factoring out 25 (that is, ) from 850, then using the fact that .

When we combine like terms, we can see that . The two terms are therefore both equal to the same value.

 is always equal to ; therefore, 5 raised to 4 times 5 raised to 5 must equal 5 raised to 9.

is always equal to . Therefore, 5 raised to 9, raised to 20 must equal 5 raised to 180.

First, multiply inside the parentheses: .

Then raise to the 7th power: .

Remember, we add exponents when their bases are multiplied, and multiply exponents when one is raised to the power of another. Negative exponents flip to the denominator (presuming they originally appear in the numerator). 

(y²)(y⁴) = y²⁺⁴ = y⁶

Plug in two different values for y.

Plug in y = 1: y⁸ = y⁶ 

Plug in y = 2: y⁸ > y⁶ 

Since the results differ, the relationship cannot be determined from the information given.