12A - Index laws
Our study of exponential and logarithmic functions begins with index laws. Consider \(a^m\), we consider \(a\) to be the base and \(m\) to be the index or power. When two terms have the same base the following index laws can be applied:
\[a^m \times a^n = a^{m+n}\] |
\[\frac{a^m}{a^n}=a^{m-n}\] |
\[(a^m)^n=a^{m \times n}\] |
\[(ab)^m=a^m b^m\] |
\[a^{-m}=\frac{1}{a^m}\] |
\[a^{\frac{p}{q}}=\sqrt[q]{a^p}\] |
\[a^0=1\] |
\[(\frac{a}{b})^m=\frac{a^m}{b^m}\] |
Important observations about bases and powers:
- If the exponent/index is the unknown, express both sides of the equation as exponents with the same base and then equate the powers. That is, if \(a^x=a^y\) then \(x=y\).
- If the base is the unknown, express both sides of the equation as exponents with the same powers and then equate the bases. That is, if \(a^x=b^x\) then \(a=b\).
12A - Example 1: Simplifying using index laws
Simplify the following expressions:
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12A - Example 1: Video solution
12A - Example 1: Practice
Question 1: Simplify the following expressions: (a) ABC (b) ABC Question 2: Simplify the following expression: \[x\] 12A - Example 1: Solutions
Question 1: ABC Question 2: ABC |
12A - Example 2: SIMPLIFYING using index laws
Simplify the following expressions
(a) \(\sqrt[3]{125}\) (b) \(9^\frac{3}{2}\) (c) \(64^\frac{5}{6}\) |
12A - Example 2: Video solutions
12A - Example 2: Practice
Question 1: Simplify the following expressions: (a) \((\sqrt[4]{81})^3\) (b) \(32^\frac{3}{5}\) (c) \(625^\frac{3}{4}\) (d) \(16^\frac{-1}{2}\) 12A - Example 2: Solutions
Question 1: (a) \((\sqrt[4]{81})^3=(3)^3=27\) (b) \(32^\frac{3}{5}=(32^\frac{1}{5})^3=(2)^3=8\) (c) \(625^\frac{3}{4}=(625^\frac{1}{4})^3=(5)^3=125\) (d) \(16^\frac{-1}{2}=(16^\frac{1}{2})^{-1}=\frac{1}{4}\) |