The Laws of Indices

The laws of indices (or exponents) are essential for simplifying expressions involving powers. Here are the fundamental laws explained step by step:

1. Product Rule: $a^p a^q = a^{p+q}$

When multiplying two powers with the same base, add the exponents.

2. Quotient Rule: $\frac{a^p}{a^q} = a^{p-q}$

When dividing two powers with the same base, subtract the exponents.

3. Power of a Power Rule: $(a^p)^q = a^{p \cdot q}$

When raising a power to another power, multiply the exponents.

4. Power of a Product Rule: $(ab)^p = a^p b^p$

A power applied to a product is distributed to each factor in the product.

5. Power of a Quotient Rule: $\left(\frac{a}{b}\right)^p = \frac{a^p}{b^p}$

A power applied to a fraction is distributed to the numerator and the denominator.

6. Zero Exponent Rule: $a^0 = 1 \quad \text{(for } a \neq 0\text{)}$

Any non-zero base raised to the power of 0 equals 1.

7. Negative Exponent Rule: $a^{-p} = \frac{1}{a^p}$

A negative exponent indicates the reciprocal of the base raised to the positive exponent.

8. Fractional Exponent Rule: $a^{\frac{1}{q}} = \sqrt[q]{a}, \quad a^{\frac{p}{q}} = (\sqrt[q]{a})^p$

A fractional exponent represents roots. For example, $a^{\frac{1}{2}} = \sqrt{a}$.

Example

Summary:

• Product Rule: Add exponents for the same base.
• Quotient Rule: Subtract exponents for the same base.
• Power of a Power: Multiply the exponents.
• Zero Exponent: Any base (except 0) raised to 0 is 1.
• Negative Exponent: Reciprocal of the base with a positive exponent.
• Fractional Exponent: Root of the base, raised to the numerator of the fraction.