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

bullet Product Rule for Exponents:

statement
For all real numbers x, and all integers m and n:    prodrule
When you are multiplying, and the bases are the same, ADD the exponents.
prodruleexp
When in doubt, expand the terms, as shown above, to see the correct answer.

expinS
1. prodex1
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. prodex3
Be sure to add only the exponents for the bases that are the SAME.
2. exprod2
Bases are the same, so the exponents are added.
Be careful when adding negative exponents.
4. prodec4
By the Distributive Property, rs is multiplied times EACH term inside the parentheses. Add the exponents of same bases in each multiplication.

 

bullet Quotient Rule for Exponents:
statement
For all real numbers x (not zero), and all integers m and n:   quotrule

When you are dividing, and the bases are the same, SUBTRACT the exponents.
quotex11
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: top1a
It is, however, possible to subtract the top exponent from the bottom exponent: top2
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.

expinS
1. quotex1
The bases are the same, so the exponents are subtracted. The coefficients (in front of the bases) are divided.
3. quotex3
Be sure to subtract only the exponents for the bases that are the SAME. Remember the implied exponent of x is 1.
2. quotex2
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. quotex4
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.


bullet Negative Exponent Rule:
statement
For all real numbers x (not zero), and all integers m:    expneg
An expression raised to a negative exponent is equal to 1 divided by the expression
with the sign of the exponent changed.

expneg11
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.

expinS Write each expression without a negative exponent:
1. neg1
3. exneg4
2. expneg2
Be careful with values not being affected by the exponent (in this case the 6).
4. expneg3
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.
switchmb


bullet Zero Exponent Rule:
statement
For all real numbers x (not zero):   x0 = 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:   ppex8a
If we tried this argument for x = 0, we would be dividing by 0, which is not possible.

expinS
1. 40 = 1
3. -20 = -1
2. (-5)0 = 1
4. 00 = undefined*
*It seems to make sense that 00 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 00 is zero? If this is not confusing enough, there are also cases where 00 yields an undefined result. For this reason, 00 is often called an indeterminate form. There is no unique universal answer as to the value of 00.



bullet Power to a Power Rule:
statement
For all real numbers x, and all integers m and n:  powrule
When you are raising a power to a power, MULTIPLY the exponents.
powrule1
When in doubt, expand the terms, as shown above, to see the correct answer.

expinS
1. powex1
Multiply the exponents.
3. powex4
Multiply the exponents. Be careful of the signs.
2. powex2
Multiply the exponents.
4. powex3
Multiply the exponents. Be careful of the signs.

 

bullet Product to a Power Rule:
statement
For all real numbers x and y, and all integers m:  pprule
This rule only works when the inside of the parentheses is a single term, a product (not addition or subtraction). Each factor of the product is raised to the new power.
pprule1
When in doubt, expand the terms, as shown above, to see the correct answer.

expinS
1. ppex11
Notice that the coefficient of 3 is also affected by the power of 2 since it is part of the product inside the parentheses.
3. ppex3
The rule still applies when working with negative exponents..
2. ppex22
Notice how the negative sign is being handled. The -1 to the power of 5 yields a negative result.
4. ppex4
Be sure to apply the exponent to the coefficient of 3.

beware
bw1
Exponents do not distribute over addition.
ppex5
bw2
Exponents do not distribute over subtraction.
ppex6


bullet Quotient to a Power Rule:
statement
For all real numbers x and y, and all integers m:  quot1
Each factor of the quotient is raised to the new power.
quot11
When in doubt, expand the terms, as shown above, to see the correct answer.

expinS
1. qex1
3. rm33
2. exq3
4. exq4



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