All About Exponents

by on October 18th, 2009

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 3^4 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.

{3^2}*{3^3} = 3^{2+3} = 3^5

{n^4}*{n^6} = n^{4+6} = n^10

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:

{4^4}/{4^2}=4^{4-2}=4^2

{y^8}/{y^3}=y^{8-3}=y^5

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:

(3^2)^3 = 3^(2*3) = 3^6

(n^4)^2 = n^(4*2) = n^8

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:

(15n)^4 = (15)^4 (n)^4

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:

15^2 +15^2 = 2(15)^2

3(15)^2 - 15^2 = 2(15)^2

Other important exponent rules

Any number raised to the power of 0 is 1:

3^0  = 1

Any number raised to the power of 1 is itself:

3^1  = 3

Zero raised to any nonzero exponent equals zero:

0^2 = 0^4 =0

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:

b^2-a^2 = (b-a)(b+a)

25-9 = 5^2-3^2= (5-3)(5+3)

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

(1/4)^2 = {1/4}*{1/4} = 1/16

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

(-3)^3 = (-3)(-3)(-3) =-27

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

(-3)^2 = (-3)(-3)=9

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

3^{-2} = 1/{3^2}  = 1/9

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 (x+72)^{1/3}=4
2) If x is the smallest integer on the list, then 1/64=x^{-2}

A) 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:

(x+72)^{1/3} = 4

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 1/64=x^{-2} then 1/64=1/{x^2} and x^2=64, 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.

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