Exponent and identity identity

Answer 1

Certainly! Here are some key formulas and properties related to exponents and identity elements:

Exponents Formulas:

1. *Product of Powers:* [ a^m \cdot a^n = a^{m+n} ] When multiplying two exponents with the same base, you add the exponents.

2. *Quotient of Powers:* [ \frac{a^m}{a^n} = a^{m-n} \quad (\text{for } a \neq 0) ] When dividing two exponents with the same base, you subtract the exponents.

3. *Power of a Power:* [ (a^m)^n = a^{m \cdot n} ] When raising an exponent to another power, you multiply the exponents.

4. *Power of a Product:* [ (ab)^n = a^n \cdot b^n ] When raising a product to a power, you raise each factor to the power.

5. *Power of a Quotient:* [ \left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} \quad (\text{for } b \neq 0) ] When raising a quotient to a power, you raise both the numerator and the denominator to the power.

6. *Zero Exponent:* [ a^0 = 1 \quad (\text{for } a \neq 0) ] Any non-zero number raised to the power of zero is 1.

7. *Negative Exponent:* [ a^{-n} = \frac{1}{a^n} \quad (\text{for } a \neq 0) ] A negative exponent indicates the reciprocal of the base raised to the opposite positive exponent.

Identity Elements:

1. *Additive Identity:* [ a + 0 = a \quad \text{and} \quad 0 + a = a ] The number 0 is the additive identity because adding 0 to any number ( a ) leaves ( a ) unchanged.

2. *Multiplicative Identity:* [ a \times 1 = a \quad \text{and} \quad 1 \times a = a ] The number 1 is the multiplicative identity because multiplying 1 by any number ( a ) leaves ( a ) unchanged.

These formulas and properties are fundamental in algebra and are used frequently in solving equations and simplifying expressions. If you need further details or examples, please let me know!