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This essential trains for: SAT-I, GMAT, AMC-8, AMC-10, Math Kangaroo 7-8, Math Kangaroo 9-10.
The number that is raised to a power is called base.
The number that is specifies the power is called exponent.
The third power of five has base 5 and exponent 3:
|Multiplying two powers of the same base:|
|Copy the base and add the exponents.|
Multiplying two powers of different bases is a possible operation, but it cannot be simplified as an expression:
|Dividing two powers of the same base:|
|Copy the base and subtract the exponents.|
We see that, if:
the exponent that results is negative!
Using integer exponents it is easy to see what the negative exponent means:
A power with some exponent is the reciprocal of the power with the opposite exponent. Examples:
All this implies that:
The zero-th power of any base is equal to 1.
Attention! A negative exponent can never make the result negative! Only a negative base may - in certain cases - make the result of the power operation negative.
Attention! The negative sign may or may not be part of the base:
Applying the definition of an integer power as repeated multiplication, we find the rule for raising a power to another power:
Since multiplication is commutative:
Often, students are puzzled by the addition/subtraction of powers. There is no rule for automatically simplifying them but there are techniques that may enable simplification in some cases.
Example 1: adding/subtracting powers of the same base.
In some cases, it may be useful to factor out the smallest power:
Example 2: adding/subtracting powers with different, but related bases:
We may be able to force the same base:
One must have the ability to observe that the bases are related. For this, it is useful to memorize the powers of 2 up to 210 and, in general, to use the prime factorizations of the integers involved.