An exponent represents repeated multiplication of a base number. The expression x^y (read as "x to the power of y") means multiplying x by itself y times. For example, 2^3 = 2 Ć 2 Ć 2 = 8.
Exponents are fundamental in mathematics and science, used in scientific notation, compound interest calculations, exponential growth/decay models, and many areas of physics and engineering.
For positive integer exponents:
For negative, fractional, or decimal exponents, more advanced techniques are used.
Exponent Laws:
Problem: Find 2 to the power of 5.
Solution:
Problem: Find 10 to the power of -2.
Solution:
Problem: Find 4 to the power of 1/2.
Solution:
0^0 is considered indeterminate in most contexts, though in some fields like combinatorics, it's defined as 1. Most calculators will either return 1 or an error for this case.
A negative exponent means "take the reciprocal." For example, x^(-n) = 1/(x^n). So 2^(-3) = 1/(2^3) = 1/8 = 0.125.
A fractional exponent represents a root. x^(1/n) means the nth root of x. For example, 8^(1/3) = ā8 = 2. The numerator represents a power, so x^(m/n) = (nāx)^m.
Exponents involve repeated multiplication of the same base (x^y = x Ć x Ć ... Ć x), while factorial involves multiplying consecutive integers (n! = n Ć (n-1) Ć ... Ć 1).