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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:

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Multiplying two powers of the same base:
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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:

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Dividing two powers of the same base:
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Copy the base and subtract the exponents.

We see that, if:

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the exponent that results is negative!

Using integer exponents it is easy to see what the negative exponent means:

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A power with some exponent is the reciprocal of the power with the opposite exponent. Examples:

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All this implies that:

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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.

If the base is positive, the power is positive regardless of the exponent.equationequation
If the base is negative, the power is negative only if the exponent is an odd integer.equationequation

Attention! The negative sign may or may not be part of the base:

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Applying the definition of an integer power as repeated multiplication, we find the rule for raising a power to another power:

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Examples:

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Since multiplication is commutative:

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Undefined operation:

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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:

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Example 2: adding/subtracting powers with different, but related bases:

We may be able to force the same base:

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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.