Exponential Operations - GRE Quantitative Reasoning

(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.

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

(23 )2 or 23*2 = 64

(22 )3 or 22*3 = 64

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

When you multiply exponents, you add the common bases:

y4 x7 + y5x6 + y18x4 + y3x26

By using exponent rules, we can simplify Quantity B.

Also, we can simplify Quantity A.

Since n is positive,

Since the base is 5 for each term, we can say 2 + n =12. Solve the equation for n by subtracting 2 from both sides to get n = 10.

Rewrite the term on the left as a product. Remember that negative exponents shift their position in a fraction (denominator to numerator).

The term on the right can be rewritten, as 27 is equal to 3 to the third power.

Exponent rules dictate that multiplying terms allows us to add their exponents, while one term raised to another allows us to multiply exponents.

We now know that the exponents must be equal, and can solve for .

First, simplify by adding the exponents to get .

Then simplify by multiplying the exponents to get .

This gives us . We cannot simplify any further.

Start by simplifying each individual term between the plus signs. We can add the exponents in and so each of those terms becomes . Then multiply the exponents in so that term also becomes . Thus, we have simplified the expression to which is .

To attempt this problem, note that .

Now note that when multiplying numbers, if the base is the same, we may add the exponents:

This can in turn be written in terms of nine as follows (recall above)

When dealing with exponenents, when multiplying two like bases together, add their exponents:

However, when an exponent appears outside of a parenthesis, or if the entire number itself is being raised by a power, multiply:

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 .

Let's look at the two parts of the multiplication separately. Remember that (–8)1/3 will be negative. Then 641/2 + (–8)1/3 = 8 – 2 = 6.

For the second part, we can cancel some exponents to make this much easier. 43/16 = 43/42 = 4. Similarly, 3171/3169 = 3171–169 = 32 = 9. So 43/16 – 3171/3169 = 4 – 9 = –5.

Together, \[641/2 + (–8)1/3\] * \[43/16 – 3171/3169\] = 6 * (–5) = –30.

The easiest way to solve this is to simplify the fraction as much as possible. We can do this by factoring out the greatest common factor of the numerator and the denominator. In this case, the GCF is .

Now, we can cancel out the from the numerator and denominator and continue simplifying the expression.

Simplifying the equation gives b6/(b3+x) = b5.

In order to satisfy this case, x must be equal to –2.

7/8 is being raised to the 5th power and to the 1/2 power at the same time. We multiply these to find n.

(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.

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

xaxb = xa+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