When negative numbers are raised to powers, the result may be positive or negative. A negative number raised to an even power is always positive, and a negative number raised to an odd power is always negative.
For example,
(−3)^{4} = −3 × −3 × −3 × −3 = 81
(−3)^{3}= −3 × −3 × −3 = −27
Take note of the parenthesis: (−3)^{2} = 9, but −3^{2} = −9
Exponents can also be negative or zero; such exponents are defined as follows.
• For all nonzero numbers a, a^{0} = 1.
• The expression 0^{0} is undefined.
• For all nonzero numbers a,
Take note that
A square root of a nonnegative number n is a number r such that r^{2} = n.
For example, 5 is a square root of 25 because 5^{2} = 25.
Another square root of 25 is −5 because (−5)^{2} is also equals to 25.
The symbol used for square root is . (The symbol is also called the radical sign).
Every positive number a has two square roots: , which is positive, and , which is negative. = 4 and − = −4
The only square root of 0 is 0. Square roots of negative numbers are not defined in the real number system.
Rules |
Examples |
Examples |
A square root is a root of order 2. Higher-order roots of a positive number n are defined similarly.
The cube root is a root of order 3.
For example,
8 has one cube root. The cube root of 8 is 2 because 2^{3} = 8.
−8 has one cube root. The cube root of −8 is −2 because (−2)^{3} = −8
The fourth root is a root of order 4.
For example,
8 has two fourth roots.
because 2^{4} = 16 and (−2)^{4} = 16
These n^{th} roots obey rules similar to the square root.
There are some notable differences between odd order roots and even-order roots (in the real number system):
• For odd-order roots, there is exactly one root for every number n, even when n is negative. For example, the cube root of 8 is 2 and the cube root of −8 is −2.
• For even-order roots, there are exactly two roots for every positive number n and no roots for any
negative number n. For example, the fourth root of 16 is 2 and −2 and there is no fourth root for −16.
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