Explanation:
The law of exponents, also known as the laws of indices, are mathematical rules that govern the behavior of exponential expressions. These laws are particularly useful when working with expressions involving powers or exponents. Here are some fundamental laws of exponents:
1. **Product Law:** When you multiply two exponential expressions with the same base, you can add their exponents. This is expressed as:
a^m * a^n = a^(m + n)
For example, 2^3 * 2^4 = 2^(3 + 4) = 2^7.
2. **Quotient Law:** When you divide two exponential expressions with the same base, you can subtract their exponents:
a^m / a^n = a^(m - n)
For example, 3^5 / 3^2 = 3^(5 - 2) = 3^3.
3. **Power Law:** When you raise an exponential expression to another exponent, you can multiply the exponents:
(a^m)^n = a^(m * n)
For example, (4^2)^3 = 4^(2 * 3) = 4^6.
4. **Negative Exponent Law:** If you have a negative exponent, you can move the base to the denominator of a fraction and change the sign of the exponent:
a^(-n) = 1 / a^n
For example, 2^(-3) = 1 / 2^3 = 1/8.
5. **Zero Exponent Law:** Any nonzero number raised to the power of zero is equal to 1:
a^0 = 1 (for a ≠ 0)
For example, 5^0 = 1.
These laws are fundamental in simplifying and manipulating expressions involving exponents, making them a key concept in algebra and mathematics in general. They are used extensively in various mathematical and scientific disciplines.