Exponents - GMAT Quantitative

If is negative, then:

Even powers and are positive, with and .

Since , ,

It follows that , and .

If , then - that is, . So changing the signs of the exponents reverses the order. As a result,

.

Odd powers and are negative, with and .

, so , and .

As before, changing their exponents to their opposites reverses the order:

Setting the negative numbers less than the positive numbers:

.

$$\frac{6^{3}$$}{36} = $$\frac{6^{3}$$$}{6^{2}$} = 6

$$\frac{3^{68}$$$}{3^{67}$} = $3^{68-67}$ = 3

Putting these together,

$$\frac{6^{3}$$}{36} + $$\frac{3^{68}$$$}{3^{67}$}= 6 + 3 = 9

$3x^{4}$times $x^{2}$ $=3x^{6}$

Then, $3x^{4}$times $x^{2}$$+x^{2}$-x = $3x^{6}$$+x^{2}$-x = $x(3x^{5}$+x-1)

$4^{$\frac{3}$${2}$}=(4^{$\frac{1}$${2}$})^{3}$ = $2^{3}$ = 8

$27^{$\frac{2}$${3}$}=(27^{$\frac{1}$${3}$})^{2}$ = $3^{2}$ = 9

Then putting them together, $4^{$\frac{3}${2}$} + $27^{$\frac{2}${3}$} = 8 + 9 = 17

We have .

So , and .

The easiest way to simplify is to work from the inside out. We should first get rid of the negatives in the exponents. Remember that variables with negative exponents are equal to the inverse of the expression with the opposite sign. For example, So using this, we simplify:

Now when we multiply variables with exponents, to combine them, we add the exponents together. For example,

Doing this to our expression we get it simplified to .

The next step is taking the inside expression and exponentiating it. When taking an exponent of a variable with an exponent, we actually multiply the exponents. For example, . The other rule we must know that is an exponent of one half is the same as taking the square root. So for the So using these rules,

The number in front must have an absolute value greater than or equal to 1, and less than 10. Of these choices, only qualifies.

We can use the fact that to see that

Since , we have

.

The left and right sides of the equation have the same base, so we can equate the exponents and solve:

First, write each expression as a base 3 logarithm:

since

Rewrite the expression accordingly, and apply the logarithm sum and difference rules:

Inspect the first few powers of 6; a pattern emerges.

As you can see, the last two digits repeat in a cycle of 5.

789 divided by 5 yields a remainder of 4; the pattern that becomes apparent in the above list is that if the exponent divided by 5 yields a remainder of 4, then the power ends in the diigts 96.

Use the properties of exponents as follows:

Use the properties of exponents as follows:

Apply the power of a power principle twice by multiplying exponents: