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The product of two powers with the same non-zero base a and integer exponents m and n can be written as a single power with base a and exponent m+n.
a^m * a^n = a^(m+n)
To prove the identity, it will be shown that the left-hand side is equivalent to the right-hand side. To do so, a^m and a^n will be written as products using the definition of a power. a^m&=a* a * ... * a_(mtimes) a^n&=a* a * ... * a_(ntimes) With this information, the left-hand side of the identity can be rewritten using multiplication. a^m * a^n = a* a * ... * a_(mtimes) * a* a * ... * a_(ntimes)_(m+ ntimes) By the definition of a power, multiplying a number by itself m+n times is equivalent to raising that number to the power of m+n. a* a * ... * a_(mtimes) * a* a * ... * a_(ntimes)_(m+ ntimes) = a^(m+ n) The Product of Powers Property has been proved.
a^m * a^n = a^(m+n)
The ratio of two powers with the same non-zero base a and integer exponents m and n can be written as a single power with base a and exponent m-n.
a^m/a^n=a^(m-n)
To prove the identity, it will be shown that the left-hand side is equivalent to the right-hand side. To do so, a^m and a^n will be written as products using the definition of a power. a^m/a^n =a * a * a * ... * a^(m times)/a * a * a * ... * a_(n times) Next, the common factors will be eliminated. It will be arbitrarily assumed that m>n. a^m/a^n =a * a * a * ... * a^(m times)/a * a * a * ... * a_(n times) Note that when the quotient is simplified, the number of remaining factors will be m-n. a^m/a^n =a* a * ... * a_(m-ntimes) 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.
a^m/a^n=a^(m-n)
Note that this property is also valid for m≤ n. If m=n, then the exponent will be zero. If m
A power with a non-zero base a and an integer exponent m that is raised to another integer exponent n can be written as a power with base a and exponent m* n.
(a^m)^n = a^(m* n)
For the rule to be true for a=0, both exponents must be greater than zero.
To prove the identity, it will be shown that the left-hand side is equivalent to the right-hand side. To do so, (a^m)^n will be rewritten as a product using the definition of a power. (a^m)^n=a^m * a^m * ⋯ * a^m_(ntimes) Then, all occurrences of the expression a^m will be written as products using the definition of a power one more time. (a^m)^n=a* ⋯ * a_(m times) * a* ⋯ * a_(m times) * ⋯ * a* ⋯ * a_(m times) ↘ ↓ ↙ n times Notice that the base a is multiplied by itself m* n times. By the definition of a power, the right-hand side of the above equation is equivalent to raising a to the power of m* n. a * a * a * ⋯ * a_(m* ntimes) =a^(m* n) The Power of a Power Property has been proved.
(a^m)^n = a^(m* n)
A power with an integer exponent m whose base is the product of two non-zero factors a and b can be written as the product of two powers with bases a and b and the same exponent m.
(ab)^m = a^m b^m
For this rule to be valid when either a or b is 0, m must be greater than zero.
To prove the identity, it will be shown that the left-hand side is equivalent to the right-hand side. To do so, (ab)^m will be written as a product using the definition of a power. (ab)^m=(ab)* (ab) * ⋯ * (ab)_(mtimes) Next, the parentheses can be removed. m times ↗ ↑ ↖ (ab)^m= a* b* a* b * ⋯ * a* b ↘ ↓ ↙ m times Now, the Commutative Property of Multiplication can be used. (ab)^m=a* a * ⋯ * a_(mtimes) * b* b * ⋯ * b_(mtimes) Finally, using the definition of a power again, the right-hand side of the equation can be written as the product of a^m and b^m.
(ab)^m = a^m b^m
The Power of a Product Property has been proven.
A power with an integer exponent m whose base is a ratio of a non-zero numerator a to a non-zero denominator b can be written as the ratio of two powers with bases a and b and the same exponent m.
(a/b)^m=a^m/b^m
For the rule to be true for a=0, m must be greater than zero.
To prove the identity, it will be shown that the left-hand side is equivalent to the right-hand side. To do so, ( ab)^m will be rewritten as a product using the definition of a power. (a/b)^m = a/b * a/b * a/b * ... * a/b_(m times) Next, the fractions will be multiplied. (a/b)^m =a * a * a * ... * a^(m times)/b * b * b * ... * b_(m times) 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 a^m to the denominator b^m.
(a/b)^m=a^m/b^m
The Power of a Quotient Property has been proven.
If a is a non-zero real number and n is a positive integer, then a raised to the power of - n is equal to 1 over a raised to the power of n.
a^(- n)=1/a^n
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.
Write as a difference
Write as a sum
Commutative Property of Addition
a^(m+n)=a^m*a^n
.LHS /a^n.=.RHS /a^n.
Any non-zero real number raised to the power of 0 is equal to 1.
a^0=1