Exponents - GRE Quantitative Reasoning

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.

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.

Let's pick numbers. For quantitative comparisons with exponents, it's good to try 0, a negative number, and a fraction.

0: 02 = 0, 03 = 0, so the two quantities are equal.

–1: (–1)2 = 1, (–1)3 = –1, so Quantity A is greater.

Already we have a contradiction so the answer cannot be determined.

First rewrite quantity B so that it has the same base as quantity A.

can be rewriten as , which is equivalent to .

Now we can compare the two quantities.

is greater than .

For exponent problems like this, the easiest thing to do is to break down all the numbers that you have into their prime factors. Begin with the number given to you:

Now, in order for you to have a number that is a multiple of this, you will need to have at least in the prime factorization of the given number. For each of the answer choices, you have:

; This is the answer.

Because the numbers involved in your fraction are so large, you are going to need to do some careful manipulating to get your answer. (A basic calculator will not work for something like this.) These sorts of questions almost always work well when you isolate the large factors and notice patterns involved. Let's first focus on the numerator. Go ahead and break apart the into its prime factors:

Note that these have a common factor of . Therefore, you can rewrite the numerator as:

Now, put this back into your fraction:

With problems like this, it is always best to break apart your values into their prime factors. Let's look at the numerator and the denominator separately:

Numerator

Continuing the simplification:

Now, these factors have in common a . Factor this out:

Denominator

This is much simpler:

Now, return to your fraction:

Cancel out the common factors of :

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.

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:

(xy)4 can be rewritten as x4y4 and z0 = 1 because a number to the zero power equals 1. After simplifying, you get 1/y.

Start by simplifying the numerator and denominator separately. In the numerator, (c3)2 is equal to c6. In the denominator, c2 * c4 equals c6 as well. Dividing the numerator by the denominator, c6/c6, gives an answer of 1, because the numerator and the denominator are the equivalent.

Distribute the outside exponents first:

Divide the coefficient by subtracting the denominator exponents from the corresponding numerator exponents:

The numerator is simplified to (by adding the exponents), then cube the result. a24/a6 can then be simplified to .