Laws of Indices

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 \(2^3\).