# All About Exponents

An exponent is a fancy way of writing the value of a number multiplied several times (up to infinity) by itself. 3 x 3 x 3 x 3 can also be written as and can be expressed as “three raised to the fourth power” or “three to the fourth”. The large number (3) is called the base and the little number (in this case 4) is called the exponent. Where the exponent is an integer, exponentiation corresponds to repeated multiplication, whereas multiplication corresponds to repeated addition.

There are several properties and rules to keep in mind with exponents:

**Multiplying numbers with the same base (Product Rule for exponents)**

When multiplying numbers with the same base, all you do is add their exponents.

**Dividing numbers with the same base (Quotient Rule for exponents)**

When dividing numbers that have the same base, subtract the bottom number’s exponent from the top number’s exponent:

**Raising a power to a power**

When you raise an exponent to another power, or raise a power to a power, you need to multiply the exponents:

**Distributing exponents**

When you see several numbers within parentheses and an exponent outside of the parentheses, you need to make sure you distribute that exponent to all of the numbers within those parentheses:

**Adding/subtracting exponents with same base and exponent**

You can add and subtract numbers with the same base and exponent, but they must have the same base and exponent:

**Other important exponent rules**

Any number raised to the power of 0 is 1:

Any number raised to the power of 1 is itself:

Zero raised to any nonzero exponent equals zero:

The difference of squares (really get to know this, as the GMAT loves to test you on this). Whenever you see an equation that can be simplified into the difference of squares, definitely factor it, it will probably help you out tremendously:

If you raise a positive fraction that is less than 1 to a power, the fraction gets smaller:

If you raise a negative number to an odd power, the number gets smaller:

If you raise a negative number to an even power, the number becomes positive:

Any number to the negative power *x* is equal to the reciprocal of the same number to the positive power *x*:

**Example Question**

A list contains 11 consecutive integers. What is the greatest integer on the list?

1) If x is the smallest integer on the list, then

2) If x is the smallest integer on the list, thenA) Statement 1 alone is sufficient but statement 2 alone is not sufficient to answer the question asked.

B) Statement 2 alone is sufficient but statement 1 alone is not sufficient to answer the question asked.

C) Both statements 1 and 2 together are sufficient to answer the question but neither statement is sufficient alone.

D) Each statement alone is sufficient to answer the question.

E) Statements 1 and 2 are not sufficient to answer the question asked and additional data is needed to answer the statements.

**Solution**

If we can determine the smallest integer on the list or a specific integer on the list when the list is written in increasing order, we can determine the greatest integer on the list.

1) Sufficient: We’re given one variable and one equation for the smallest integer on the list. That means we could solve for smallest integer and add 10 to find the greatest integer. If you don’t see this, consider:

Cubing both sides, *x* + 72 = 64. Then *x* + 72 = 64 and *x* = -8. Adding 10 to -8, the greatest integer is 2. Eliminate choices B, C and E.

2) Insufficient: If then and , so *x* could be -8 or 8.

There are two different possibilities for the smallest integer on the list, so there must be two different possibilities for the greatest integer on the list. Statement 2) is insufficient, leaving the correct answer choice as A.

Have a question on exponents? Please post them in the comment box below.

## 5 comments

David Jenkins on October 19th, 2009 at 2:30 am

Thanks for this.

I have seen a problem that came up with 0^0 and I have seen two different answers?

See: http://mathforum.org/dr.math/faq/faq.0.to.0.power.html http://www.platinumgmat.com/gmat_study_guide/exponential_equations

Martin on October 20th, 2009 at 10:38 pm

David,

As the first link says (as well as the article), it is generally the best rule to follow that any number, including 0, raised to the zero, is 1. Don't get this confused with zero raised to any nonzero exponent is zero.

hope this helps.

Charlie on October 19th, 2009 at 3:30 pm

Can also perform operations (multiplication and division) on different bases that have same exponent. Perform the operation and keep the exponent.

4^2 * 2^2 = 8^2.

4^2 = 2 x 2 x 2 x 2

2^2 = 2 x 2

2x2x2x2x2x2 = 8 x 8 = 8^2

haseen on October 16th, 2010 at 12:15 pm

what are applications of laws? i mean tell all about applications of laws.

blah on June 2nd, 2012 at 7:42 am

hi this doesnt help at all