The following examples come from RegentsPrep.org
Practice Problems
Basic Exponents
Applied Problems
Exponents are the mathematician’s shorthand.


In general, the format for using exponents is:
(base)^{exponent}
where the exponent tells you how many of the base are being multiplied together.

Consider: 2 2 2 is the same as 2^{3}, since there are
three 2’s being multiplied together.
Likewise, 5 5 5 5 = 5^{4}, because there are
four 5’s being multiplied together.
Exponents are also referred to as “powers”.
For example, 2^{3} can be read as “two cubed” or as “two raised to the third power”.
Exponents of Negative Values

When we multiply negative numbers together, we must utilize parentheses to switch to exponent notation.
(3)(3)(3)(3)(3)(3) = (3)^{6}
BEWARE!! 3^{6} is NOT the same as (3)^{6}
The missing parentheses mean that 3^{6} will multiply six 3’s together first (by order of operations), and then take the negative of that answer.
(3)^{6} = 729 but 3^{6} = 729
so be careful with negative values and exponents !
Note: Even powers of negative numbers allow for the negative values to be arranged in pairs. This pairing guarantees that the answer will always be positive.
(5)^{6} 
= (5)•(5) • (5)•(5) • (5)•(5) ← All pairs. 

= 25 • 25 • 25 

= 15625 (a positive answer) 
Odd powers of negative numbers, however, always leave one factor of the negative number not paired. This one lone negative term guarantees that the answer will always be negative.
(5)^{5} 
= (5)•(5) • (5)•(5) • (5) ←One lone, unpaired, negative. 

= 25 • 25 • (5) 

= 3125 (a negative answer) 

The number zero may be used as an exponent.
The value of any expression raised to the zero power is 1.
(Except zero raised to the zero power is undefined.)
Base^{0}

Value

2^{0 } =

1

(6)^{0 } =

1

4^{0 }=

1

8^{0} =

1
Raise to the zero power first: 8^{0}=1
then take the negative.

0^{0} =

undefined

Negative numbers as exponents have a special meaning.
The rule is as follows:
base^{ negative exponent }=



For example:
Negative Exponent

Positive Exponent

4^{1 }=


7^{3 }=


(5)^{2 }=


When working with units and exponents (or powers), remember to adjust the units appropriately.
(36 ft)^{3 }

= (36 ft) • (36 ft) • (36 ft) 

= (36 • 36 • 36) (ft • ft • ft) 

= 46656 ft^{3} 
Rule:
For all numbers x and all integers m and n ,

“This simply means … 
when you are multiplying,
and the bases are the same,
you ADD the exponents.”


Consider:

Observe this rule at work in the following examples:


1.


The bases are the same (all 2’s), so the exponents are added.


2.


The bases are the same, so the exponents are added. Notice how the numbers in front of the bases (7 and 1) are being multiplied.


3.


The bases are the same (all a’s), so the exponents are added.


4.


The bases are the same (all x‘s), so the exponents are added.
Be careful when adding the negative exponent.


5.


The bases are the same, so the exponents are added.
The numbers in front of the bases are multiplied.


6.


The exponents are added for the bases that are the SAME. The numbers in front, the coefficients, are multiplied. Don’t forget powers of 1, such as the power associated with t.


7.


The exponents are added for the bases that are the SAME. The coefficients are multiplied.


8.


The 9x is multiplied times EACH term inside the parentheses, adding the exponents as the multiplication occurs.


9.


The ab is multiplied times EACH term inside the parentheses, adding the exponents of similar bases as this process occurs.


10.


The x^{2}y is multiplied times EACH term inside the parentheses, adding the exponents of similar bases as this process occurs.


Take one more look at the distributive property at work with a set of parentheses, along with this new rule:
Use the distributive property to simplify:

Rule:
For all numbers x (not zero) and all integers m and n ,

“This simply means …
when you are dividing,
and the bases are the 
same, you SUBTRACT the exponents.”
(top exponent subtract bottom exponent)


Consider:. ..when in doubt, expand terms

Observe this rule at work in the following examples:


1.


The bases are the same (both 2’s), so the exponents are subtracted.


2.


The bases are the same, so the exponents are subtracted. Notice how the numbers in front of the bases (7 and 1) are being divided.


3.


The bases are the same (both a‘s), so the exponents are subtracted.
Remember: top exponent minus bottom exponent.


4.


The bases are the same, so the exponents are subtracted.
The numbers in front of the bases are divided.


5.


The exponents are subtracted for the bases that are the SAME. The numbers in front, the coefficients, are divided.



6.


Notice what happened to the bases with the same exponents — they reduced to the number 1.


7.


Remember: top exponent minus bottom exponent.
Remember: negative exponents can be written as a fraction.


Rule:
For all numbers x, and integers n and m,

“This simply means … 
when raising a power to a power, multiply the exponents.”


Consider:

Take a look at the following examples which illustrate this rule:
Rule:
For all numbers x and y , and integers n,

“Notice: each factor of the product gets raised to the new power.”

“Be sure to notice that this rule ONLY works when the inside of the parentheses is a single term (a product). ”
“(no + signs or – signs separating the items).”


Consider:

Check out these examples of this rule at work:

… notice how the 2 was also affected by the power of 3 since it was inside the parentheses.

Notice that the 2 is not affected by the power of 3 since it is not within the parentheses.

Remember that 1 to a power of 3 is 1.

Be careful when working with negative exponents.

Be sure to cube the 2 value.

Did you notice the subtraction in this problem?

Formulas often involve working with powers.

Additional Resources
Rules of Exponents
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