Exponents - GRE Quantitative Reasoning

Begin by expanding Quantity A:

Now in order to compare this to Quantity B:

A good method would be to subtract shared terms from each Quantity; in this case, both quantities have an and term. Removing them gives:

Quantity A' :

Quantity B' :

The question now is the sign of Quantity A'; if it's always positive, Quantity A is greater. If it's always negative, Quantity B is greater. If it is zero, the two are the same.

We know that

Now compare and :

Looking at absolute values so that we're only considering positive terms:

From this it follows that by multiplying across the inequality :

From this we can determine that the magnitude of is greater. However, since this is the product of one negative number and two positive numbers, is negative, and the sum of and must in turn be negative, and so Quantity A' must be negative!

From this we can say that Quantity B is greater.

use FOIL to factor the expression.

First: (x3)(x) = x4

Outside (x3)(7) = 7x3

Inside (–3)(x) = –3x (Don't forget the negatives!)

Last (3)(7) = –21

The difference of squares formula says (x + a)(x - a) = x2 - a2.

Thus, Quantity A equals 8.

Therefore, Quantity B is greater.

Use the foil method: (3x - 3) (x + 7) = 3x2 +21x - 3x - 21 = 3x2 +18x -21 so t = 18.

Quantity A: 22 + 32 = 4 + 9 = 13

Quantity B: (2 + 3)2 = 52 = 25, so Quantity B is greater.

We can also think of this in more general terms. _x_2 + _y_2 does not generally equal (x + y)2.

Use the method of FOIL (First, Outside, Inside, Last) and add exponents for like bases:

To approach this problem, consider the two quantities

Quantity A:

Quantity B:

They are in different forms, so expand quantity A:

Quantity A:

Quantity B:

Now, for the purpose of comparison, subtract shared terms from each quantity:

Quantity A*:

Quantity B*:

Both and are negative, non-zero values. Since is a product of two negative values, it must be positive. Quantity B must be greater than Quantity A.

This problem is deceptive. Looking at Quantity A, one may think to factor and reduce it as follows:

Which is identical to Quantity B.

However, we cannot ignore that in the original fraction! We are given no conditions as to the value of . If , then Quantity A would be undefined. Since we're not given the condition , we cannot ignore this possibility.

The relationship cannot be established.

You must FOIL the expression which means to multiply the first terms together followed by the outer terms, then the inner terms and lastly, the last terms.

The expression written out looks like

.

You multiple both First terms to get .

Then the outer terms are multiplied .

Then you multiple the inner terms together .

Finally you multiply the last terms of each .

This gives you or .

Divide the coefficients and subtract the exponents.

For this problem we need to convert meters into kilometers and seconds into hours. Therefore we get,

Multiplying this out we get

100 miles = 528,000 feet. To put a number in scientific notation, we put a decimal point to the right of our first number, giving us 5.28. We then multiply by 10 to whatever power necessary to make our decimal equal the value we are looking for. For 5.28 to equal 528,000 we must multiply by 10^5.

Therefore, our final answer becomes:

This question requires you to have an understanding of scientific notation. Begin by multiplying the two numbers:

To use scientific notation, the number to the left of the decimal has to be between 1 and 10. In this case, we are looking to move the decimal place until we are left with 9 on the left of the decimal. Count the number of places that the decimal will have to move. In this case, it is five. Therefore:

Note: The notation is raised to a negative power because we moved the decimal from left to right.

First, break the problem into two segments: the amount Jack invests in the high-yield savings, and the amount Jack invests in the simple interest account (10,000 and 5,000 respectively).

Now let's work with the high-yield savings account. $10,000 is invested at an annual rate of 8%, compounded quarterly. We can use the compound interest formula to solve:

Plug in the values given:

Therefore, Jack makes $824.32 off his high-yield savings account. Now let's calculate the other interest:

Add the two together, and we see that Jack makes a total of, off of his investments.

Let's all convert the bases to .

This one may be intimidating but .

Therefore,

is not like the answers so this is the correct answer.

Each year, you can calculate your interest by multiplying the principle () by . For one year, this would be:

For two years, it would be:

, which is the same as

Therefore, you can solve for a five year period by doing:

Using your calculator, you can expand the into a series of multiplications. This gives you , which is closest to .

It is easiest to break this down into steps. For each year, you will multiply by to calculate the new value. Therefore, let's make a chart:

After year 1: ; Total interest:

After year 2: ; Let us round this to ; Total interest:

After year 3: ; Let us round this to ; Total interest:

Thus, the positive difference of the interest from the last period and the interest from the first period is:

This problem is quite simple if you recall that the units place of powers of 2 follows a simple 4-step sequence.

Observe the first few powers of 2:

21 = 2, 22 = 4, 23 = 8, 24 = 16, 25 = 32, 26 = 64, 27 = 128, 28 = 256 . . .

The units place follows a sequence of 2, 4, 8, 6, 2, 4, 8, 6, etc. Thus, divide 102 by 4. This gives a remainder of 2.

The second number in the sequence is 4, so the answer is 4.

We know that the expression must be negative. Therefore one or all of the terms x7, y8 and z10 must be negative; however, even powers always produce positive numbers, so y8 and z10 will both be positive. Odd powers can produce both negative and positive numbers, depending on whether the base term is negative or positive. In this case, x7 must be negative, so x must be negative. Thus, the answer is x < 0.