Properties of Exponents
An exponent is a simplified notation for writing down powers of numbers. For example, 2 * 2 * 2 * 2 can be written with an exponent: 2 * 2 * 2 * 2 = 2^4, which is read "two to the fourth power." Two well-known exponents are: 2 * 2 = 2^2 or "two to the second power," commonly "two squared," and 2 * 2 * 2 = 2^3, "two to the third power," also known as "two cubed."
As with most mathematical notation, there are specific properties to follow when using exponents.
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Taking an Exponent to a Power
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To take a number that already includes an exponent to another power, multiply the two exponents together: (x^m)^n = x^(m*n).
Taking a Number to a Negative Power
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When taking a number to a negative power, use the reciprocal of that number (i.e., put that number and it's power without the negative sign in the denominator of the fraction):
x^(-1) = 1/(x^n). -
Multiplication of Numbers with Exponents
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To multiply two numbers with different exponents, you simply add the exponents:
x^2 * x^5 = x^(2+5) = x^7.When multiplying two numbers in parentheses taken to a particular power, take the separate numbers to the power first, then multiply them together:
(x * y)^m = x^m * y^m.
Division of Numbers with Exponents
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To divide two numbers with different exponents, subtract the exponents:
x^5/x^2 = x^(5-2) = x^3.When dividing two numbers in parentheses and then taking the result to the specified power, take the exponent of the two numbers first, then solve the division:
(x/y)^n = x^n/y^n.When dividing two numbers in parentheses and then taking the result to the specified negative power, flip the fraction to get a positive exponent:
(x/y)^(-n) = (y/x)^n.Keep in mind that neither x nor y can be zero in the above cases.
Special Cases
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A number to the first power (x^1) is itself, i.e., x^1 = x.
A number to the zeroth power (x^0) is defined to be 1, i.e., x^0 = 1.
Zero to the zeroth power (0^0) is defined as 1, i.e., 0^0 = 1. Keep in mind that this is a mathematical definition not a mathematical truth.
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