Potenzgesetzen

Potenzgesetzen, also known as the laws of exponents, are a fundamental set of rules in mathematics that govern the manipulation of powers. These laws simplify calculations and expressions involving exponents.

The first law states that when multiplying powers with the same base, you add the exponents: a^m a^n = a^(m+n). For example, x^2 x^3 = x^(2+3) = x^5.

The second law deals with dividing powers with the same base. In this case, you subtract the exponents: a^m / a^n = a^(m-n). For instance, y^7 / y^4 = y^(7-4) = y^3.

The third law concerns raising a power to another power. Here, you multiply the exponents: (a^m)^n = a^(mn). An example is (b^2)^5 = b^(25) = b^10.

The fourth law states that when raising a product to a power, you raise each factor to that power: (ab)^n = a^n b^n. For example, (xy)^3 = x^3 y^3.

The fifth law applies to raising a quotient to a power: (a/b)^n = a^n / b^n, provided b is not zero. For instance, (c/d)^2 = c^2 / d^2.

Additionally, any non-zero number raised to the power of zero equals one: a^0 = 1. For example, 5^0 = 1. A non-zero number raised to a negative exponent is equal to the reciprocal of the number raised to the positive exponent: a^-n = 1/a^n. For example, 2^-3 = 1/2^3 = 1/8. These Potenzgesetzen are essential for algebraic manipulation and various areas of mathematics.

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