Question

Exponent properties

Answer 1

Exponent properties, also known as the laws of exponents, are rules that simplify the manipulation of expressions involving exponential terms. Here are the key properties:

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

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

3. **Power of a Power Rule**:
\[
(a^m)^n = a^{m \times n}
\]
When you raise an exponential expression to another exponent, you multiply the exponents.

4. **Power of a Product Rule**:
\[
(ab)^n = a^n b^n
\]
When you raise a product to an exponent, you can distribute the exponent to each factor in the product.

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

6. **Zero Exponent Rule**:
\[
a^0 = 1 \quad (a \neq 0)
\]
Any non-zero base raised to the power of zero is equal to one.

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

8. **Fractional Exponents**:
\[
a^{\frac{m}{n}} = \sqrt[n]{a^m}
\]
A fractional exponent indicates the root of the base, where the numerator is the power and the denominator is the root.

These properties can be used to simplify expressions and solve equations involving exponents.