
This page shows ALL of the Exponent Rules with explanations and examples.
If you want a "quick" version, see Exponent Rules Quick View.
Product Rule for Exponents:

For all real numbers x, and all integers m and n: 
When you are multiplying, and the bases are the same, ADD the exponents.
When in doubt, expand the terms, as shown above, to see the correct answer.
1.
Bases are the same, so the exponents are added.
The coefficients of 3 and 1 (numbers in front of the bases) are being multiplied. 
3.
Be sure to add only the exponents for the bases that are the SAME. 
2.
Bases are the same, so the exponents are added.
Be careful when adding negative exponents. 
4.
By the Distributive Property, rs is multiplied times EACH term inside the parentheses. Add the exponents of same bases in each multiplication. 
Quotient Rule for Exponents:

For all real numbers x (not zero), and all integers m and n: 
When you are dividing, and the bases are the same, SUBTRACT the exponents.
When in doubt, expand the terms, as shown above, to see the correct answer.
It is customary to subtract the bottom exponent from the top exponent:
It is, however, possible to subtract the top exponent from the bottom exponent:
Hint: Choose to start the subtraction with the larger of the two exponents, as this will prevent the answer from containing a negative exponent which would need further work to be simplified.
1.
The bases are the same, so the exponents are subtracted. The coefficients (in front of the bases) are divided. 
3.
Be sure to subtract only the exponents for the bases that are the SAME. Remember the implied exponent of x is 1. 
2.
Bases are the same, so the exponents are subtracted.
Notice how the top exponent minus bottom exponent yields a negative exponent answer, while the bottom exponent minus the top exponent does not. 
4.
Notice what happened to the variables "c". They reduced to the value 1. It could also be said that the subtraction of the exponents resulted in an exponent of zero. 
Negative Exponent Rule:

For all real numbers x (not zero), and all integers m: 
An expression raised to a negative exponent is equal to 1 divided by the expression
with the sign of the exponent changed.
When in doubt, expand the terms, as shown above, to see the correct answer.
Generally, we do not leave exponential expressions with negative exponents.
When asked to simplify an exponential expression, write the answer without negative exponents.
Write each expression without a negative exponent:
HINT: When working with negative exponents, the negative exponent is telling you that the factor is on the wrong side of the fraction bar. Simply move that factor to the other side of the fraction bar, and remove the negative sign from the exponent. 

Zero Exponent Rule:

For all real numbers x (not zero): x^{0} = 1 
Any nonzero real number with an exponent of 0 equals 1.
Let's see how, if we start with 1, we can end up with an exponent of zero:
For x ≠ 0:
If we tried this argument for x = 0, we would be dividing by 0, which is not possible.
1. 4^{0} = 1

3. 2^{0} = 1

2. (5)^{0} = 1

4. 0^{0} = undefined*

*It seems to make sense that 0^{0} should follow this rule and equal "one". But when you consider that 0 to any exponent (other than zero) is zero, could it be that 0^{0} is zero? If this is not confusing enough, there are also cases where 0^{0} yields an undefined result. For this reason, 0^{0} is often called an indeterminate form. There is no unique universal answer as to the value of 0^{0}.
Power to a Power Rule:

For all real numbers x, and all integers m and n: 
When you are raising a power to a power, MULTIPLY the exponents.
When in doubt, expand the terms, as shown above, to see the correct answer.
1.
Multiply the exponents. 
3.
Multiply the exponents. Be careful of the signs. 
2.
Multiply the exponents. 
4.
Multiply the exponents. Be careful of the signs. 
Product to a Power Rule:

For all real numbers x and y, and all integers m: 
Quotient to a Power Rule:

For all real numbers x and y, and all integers m: 
Each factor of the quotient is raised to the new power.
When in doubt, expand the terms, as shown above, to see the correct answer.
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