ALGEBRA 1 PROPERTIES OF EXPONENTS
Name of Property
Example
Explanation
Zero Exponent Property x0 =
1
(x
≠
0)
Any number (except 0) with an exponent of 0 equals 1.
Negative Exponent Property x−n =
1
xn
(x
≠
0)
Any number raised to a negative power is equivalent to the reciprocal of the positive exponent of the number.
Product of Powers Property xn•xm = xn+m (x
≠
0)
To multiply two powers with the same base, add the exponents.
Quotient of Powers Property xn xm
=
xn−m
(x
≠
0)
To divide two powers with the same base, subtract the exponents.
Power of a Product Property
(xy)n=xn•yn
(x and y ≠; 0)
To find a power of a product, find the power of each factor and then multiply.
Power of a Quotient Property
(
x y )n= xn yn
(x and y ≠; 0)
To find a power of a quotient, find the power of each part of the quotient, and then divide by canceling common factors.
Power of a Power Property
(xa)b
= xa•b (x
≠
0)
To find a power of a power, multiply the exponents.
Rational Exponent Property x 1 n = n√x x m n
=
n√xm
(x
≠
0)
Fractional powers, where a number is raised to a fraction, can be converted to a radical. The numerator becomes the exponent, and the denominator becomes the index of the radical
A rational exponent is an exponent expressed as a fraction. These expressions will convert to radical expressions.
Remember that “radical” is the name of the square root symbol
The n in this expression is the index. When it is a variable like this, it is common to say the nth root. If the index were 4, it would be the fourth root. If there is no index shown, then it is understood that the index is 2, as in a standard square root.
The index can be found in the denominator of a rational exponent.
The x is the radicand. The radicand is the expression underneath the radical symbol. It can include an exponent, but it does not have to.
The m is the exponent for the radicand. The exponent can be found in the numerator of a rational exponent. If there is no exponent on the radicand present, it is understood that it
Example
Explanation
Zero Exponent Property x0 =
1
(x
≠
0)
Any number (except 0) with an exponent of 0 equals 1.
Negative Exponent Property x−n =
1
xn
(x
≠
0)
Any number raised to a negative power is equivalent to the reciprocal of the positive exponent of the number.
Product of Powers Property xn•xm = xn+m (x
≠
0)
To multiply two powers with the same base, add the exponents.
Quotient of Powers Property xn xm
=
xn−m
(x
≠
0)
To divide two powers with the same base, subtract the exponents.
Power of a Product Property
(xy)n=xn•yn
(x and y ≠; 0)
To find a power of a product, find the power of each factor and then multiply.
Power of a Quotient Property
(
x y )n= xn yn
(x and y ≠; 0)
To find a power of a quotient, find the power of each part of the quotient, and then divide by canceling common factors.
Power of a Power Property
(xa)b
= xa•b (x
≠
0)
To find a power of a power, multiply the exponents.
Rational Exponent Property x 1 n = n√x x m n
=
n√xm
(x
≠
0)
Fractional powers, where a number is raised to a fraction, can be converted to a radical. The numerator becomes the exponent, and the denominator becomes the index of the radical
A rational exponent is an exponent expressed as a fraction. These expressions will convert to radical expressions.
Remember that “radical” is the name of the square root symbol
The n in this expression is the index. When it is a variable like this, it is common to say the nth root. If the index were 4, it would be the fourth root. If there is no index shown, then it is understood that the index is 2, as in a standard square root.
The index can be found in the denominator of a rational exponent.
The x is the radicand. The radicand is the expression underneath the radical symbol. It can include an exponent, but it does not have to.
The m is the exponent for the radicand. The exponent can be found in the numerator of a rational exponent. If there is no exponent on the radicand present, it is understood that it