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This essential trains for: SAT-I, GMAT, AMC-8, AMC-10, Math Kangaroo 7-8, Math Kangaroo 9-10.

Definitions:

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:

Examples:

Since multiplication is commutative:

Undefined operation:

Simplifying Expressions:

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.