Laws of Exponents¶

The laws of exponents (also known as the rules of exponents or indices) describe how to handle operations involving powers of the same base. Here’s a summary of the key exponent laws:

1. Product of Powers¶

When multiplying two expressions with the same base, add the exponents: \( a^m \times a^n = a^{m+n} \)

Example: \( x^2 \times x^3 = x^{2+3} = x^5 \)

2. Quotient of Powers¶

When dividing two expressions with the same base, subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \)

Example: \( \frac{x^5}{x^2} = x^{5-2} = x^3 \)

3. Power of a Power¶

When raising a power to another power, multiply the exponents: \( (a^m)^n = a^{m \times n} \)

Example: \( (x^2)^3 = x^{2 \times 3} = x^6 \)

4. Power of a Product¶

When raising a product to a power, distribute the exponent to each factor: \( (ab)^m = a^m \times b^m \)

Example: \( (2x)^3 = 2^3 \times x^3 = 8x^3 \)

5. Power of a Quotient¶

When raising a quotient to a power, distribute the exponent to both the numerator and the denominator: \( \left( \frac{a}{b} \right)^m = \frac{a^m}{b^m} \)

Example: \( \left( \frac{x}{y} \right)^2 = \frac{x^2}{y^2} \)

6. Zero Exponent Rule¶

Any non-zero base raised to the power of zero equals 1: \( a^0 = 1 \quad (\text{for } a \neq 0) \)

Example: \( 5^0 = 1 \)

7. Negative Exponent Rule¶

A negative exponent means to take the reciprocal of the base and make the exponent positive: \( a^{-m} = \frac{1}{a^m} \)

Example: \( x^{-2} = \frac{1}{x^2} \)

8. Fractional Exponent Rule¶

A fractional exponent indicates a root. The denominator of the fraction is the root, and the numerator is the power: \( a^{\frac{m}{n}} = \sqrt[n]{a^m} = \left( \sqrt[n]{a} \right)^m \)

Example: \( x^{\frac{1}{2}} = \sqrt{x} \)

These rules allow for simplifying and manipulating expressions involving exponents in algebraic operations.