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Reference

Properties of Exponents

Rule

Product of Powers Property

The product of two powers with the same non-zero base and integer exponents and can be written as a single power with base and exponent

Proof

To prove the identity, it will be shown that the left-hand side is equivalent to the right-hand side. To do so, and will be written as products using the definition of a power.
With this information, the left-hand side of the identity can be rewritten using multiplication.
By the definition of a power, multiplying a number by itself times is equivalent to raising that number to the power of
The Product of Powers Property has been proved.

Rule

Quotient of Powers Property

The ratio of two powers with the same non-zero base and integer exponents and can be written as a single power with base and exponent

Proof

To prove the identity, it will be shown that the left-hand side is equivalent to the right-hand side. To do so, and will be written as products using the definition of a power.
Next, the common factors will be eliminated. It will be arbitrarily assumed that
Note that when the quotient is simplified, the number of remaining factors will be
Finally, the definition of a power will be used one more time to express the right-hand side of the above equation as a single power. With this last step, the Quotient of Powers Property is proven.

Note that this property is also valid for If then the exponent will be zero. If the exponent will be negative.

Rule

Power of a Power Property

A power with a non-zero base and an integer exponent that is raised to another integer exponent can be written as a power with base and exponent

For the rule to be true for both exponents must be greater than zero.

Proof

To prove the identity, it will be shown that the left-hand side is equivalent to the right-hand side. To do so, will be rewritten as a product using the definition of a power.
Then, all occurrences of the expression will be written as products using the definition of a power one more time.
Notice that the base is multiplied by itself times. By the definition of a power, the right-hand side of the above equation is equivalent to raising to the power of
The Power of a Power Property has been proved.

Rule

Power of a Product Property

A power with an integer exponent whose base is the product of two non-zero factors and can be written as the product of two powers with bases and and the same exponent

For this rule to be valid when either or is must be greater than zero.

Proof

To prove the identity, it will be shown that the left-hand side is equivalent to the right-hand side. To do so, will be written as a product using the definition of a power.
Next, the parentheses can be removed.
Now, the Commutative Property of Multiplication can be used.
Finally, using the definition of a power again, the right-hand side of the equation can be written as the product of and

The Power of a Product Property has been proven.

Rule

Power of a Quotient Property

A power with an integer exponent whose base is a ratio of a non-zero numerator to a non-zero denominator can be written as the ratio of two powers with bases and and the same exponent

For the rule to be true for must be greater than zero.

Proof

To prove the identity, it will be shown that the left-hand side is equivalent to the right-hand side. To do so, will be rewritten as a product using the definition of a power.
Next, the fractions will be multiplied.
Finally, using the definition of a power one more time, the right-hand side of the equation can be written as a ratio of the numerator to the denominator

The Power of a Quotient Property has been proven.

Rule

Negative Exponent Property

If is a non-zero real number and is a positive integer, then raised to the power of is equal to over raised to the power of

It is important to note that the sign of the exponent changes, meaning that the resulting fraction is not simply the reciprocal of the original power.

Proof

To prove this identity, the Zero Exponent Property will be considered.
In the above equation, will be rewritten as and, after some algebraic manipulation and using the Product of Powers Property, the desired result will be obtained.

Rule

Zero Exponent Property

Any non-zero real number raised to the power of is equal to

Proof

To prove this identity, it will be shown that the left-hand side is always equal to for any real number To do so, the exponent will be written as a difference, for instance Then, the expression will be rearranged using the Quotient of Powers Property.

Rewrite as