U12 Laws of Integral Indices (I) Essential Formulas

This section covers the fundamental laws for manipulating expressions with integral indices. Mastery of these rules is crucial for simplifying algebraic expressions and solving exponential equations.

1 Fundamental Laws of Indices

Product of Powers

When multiplying two powers with the same non-zero base $a$, add their indices.

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

Quotient of Powers

When dividing two powers with the same non-zero base $a$, subtract the indices.

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

Power of a Power

To raise a power to another power, multiply the indices.

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

2 Special Cases and Definitions

Zero Index

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

$$ a^0 = 1 \quad (a \neq 0) $$

Negative Integral Index

A negative index represents the reciprocal of the base raised to the positive index.

$$ a^{-n} = \frac{1}{a^n} \quad (a \neq 0) $$

3 Power of a Product and Quotient

Power of a Product

When a product is raised to a power, each factor is raised to that power.

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

Power of a Quotient

When a quotient is raised to a power, both the numerator and denominator are raised to that power.

$$ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \quad (b \neq 0) $$

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