Laws of Indices

The laws of indices, also known as the rules of exponents, are essential algebraic rules used to simplify expressions involving powers of the same base. Here are the key laws:

1. Product of Powers

$$ a^m \cdot a^n = a^{m+n} $$

When you multiply two powers with the same base, add the exponents.

2. Quotient of Powers

$$ \frac{a^m}{a^n} = a^{m-n} $$

When you divide two powers with the same base, subtract the exponent of the denominator from the exponent of the numerator.

3. Power of a Power

$$ (a^m)^n = a^{m \cdot n} $$

When you raise a power to another power, multiply the exponents.

4. Power of a Product

$$ (ab)^n = a^n \cdot b^n $$

When you raise a product to a power, raise each factor in the product to the power.

5. Power of a Quotient

$$ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} $$

When you raise a quotient to a power, 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 base raised to the power of zero is equal to one.

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 positive exponent.

8. Fractional Exponent

$$ a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m $$

A fractional exponent denotes a root; the numerator is the power and the denominator is the root.

Understanding and applying these laws are crucial for manipulating and simplifying expressions involving exponents. Keep this guide handy as a quick reference!