## Laws of Indices

 4.1 Law of Indices

 Basic Laws of Indices Product Law: $$a^m\times a^n=a^{m+n}$$ Quotient Law: $$\dfrac{a^m}{a^n}=a^{m-n}$$ (for $$a\neq 0$$) Power Law: $$(a^m)^n=a^{mn}$$ Zero Exponent: $$a^0=1$$ (for $$a\neq 0$$) Negative Exponent: $$a^{-n}=\dfrac{1}{a^n}$$ (for $$a\neq 0$$)

 Fractional Indices $$a^{\frac{1}{n}}$$ represent the $$n$$-th root of $$a:a^{\frac{1}{n}}=\sqrt[n]{a}$$ $$a^\frac{m}{n}=\sqrt[n]{a^m}=(\sqrt[n]{a})^m$$

 Simplifying Expressions Combine same terms using the laws of indices. Simplify expressions with indices by applying the appropriate laws.

 Solving Equation with Indices Same base: If $$a^x=a^y$$, then $$x=y$$. Different base: Use logarithms to solve equations where the bases are different.

 Important Concepts Base: The number that is multiplied. Exponent: The number of times the base is multiplied by itself. Index Form: Expression written with exponent such as $$2^3$$.

## Laws of Indices

 4.1 Law of Indices

 Basic Laws of Indices Product Law: $$a^m\times a^n=a^{m+n}$$ Quotient Law: $$\dfrac{a^m}{a^n}=a^{m-n}$$ (for $$a\neq 0$$) Power Law: $$(a^m)^n=a^{mn}$$ Zero Exponent: $$a^0=1$$ (for $$a\neq 0$$) Negative Exponent: $$a^{-n}=\dfrac{1}{a^n}$$ (for $$a\neq 0$$)

 Fractional Indices $$a^{\frac{1}{n}}$$ represent the $$n$$-th root of $$a:a^{\frac{1}{n}}=\sqrt[n]{a}$$ $$a^\frac{m}{n}=\sqrt[n]{a^m}=(\sqrt[n]{a})^m$$

 Simplifying Expressions Combine same terms using the laws of indices. Simplify expressions with indices by applying the appropriate laws.

 Solving Equation with Indices Same base: If $$a^x=a^y$$, then $$x=y$$. Different base: Use logarithms to solve equations where the bases are different.

 Important Concepts Base: The number that is multiplied. Exponent: The number of times the base is multiplied by itself. Index Form: Expression written with exponent such as $$2^3$$.