Index Laws
Index laws are worth committing to memory, as they are very useful for simplifying expressions and solving equations algebraically. They define how to manipulate powers or indices, while keeping the mathematical integrity of equations intact.
- $$x^ax^b=x^{a+b}$$
- $$(x^a)^b=x^{ab}$$
- $$x^{-a}=\frac{1}{x^a}$$
- $$\frac{x^a}{x^b}=x^{a-b}$$
- $$x^0=1$$
- $$x^{\frac{1}{a}}=\sqrt[a]{x}$$
Index Laws with Examples
1)$$x^ax^b=x^{a+b}$$
$$x^5\; ×\; x^6\;=\;x^{5+6}\; =\;x^{11}$$
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$$x^4 ×x^8\;=\; x^{4+8}\; =\;x^{12}$$$$An\;extra \;step\;is\;included$$$$\;to\;show\;the \;working$$
2)$$(x^a)^b=x^{ab}$$
$$(x^2)^3=x^{2×3}=x^6$$
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$$(x^4)^3=x^{4×3}=x^{12}$$$$An\;extra \;step\;is\;included$$$$\;to\;show\;the \;working$$
3)$$x^{-a}=\frac{1}{x^a}$$
$$x^{-3}=\frac{1}{x^3}$$
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$$x^{-2}=\frac{1}{x^2}$$
4)$$\frac{x^a}{x^b}=x^{a-b}$$
$$\frac{x^4}{x^2}=x^{4-2}=x^2$$
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$$\frac{x^6}{x^4}=x^{6-4}=x^2$$$$Extra \;steps\;are\;included$$$$\;to\;show\;the \;working$$
5)$$x^0=1$$
$$x^{12}\;÷\;x^{12}\;=\;x^0=1$$
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$$x^9\;÷\;x^9\;=\;x^{9-9}\;=\;x^0\;=\;1$$$$Extra \;steps\;are\;included$$$$\;to\;show\;the \;working$$
6)$$x^{\frac{1}{a}}=\sqrt[a]{x}$$
$$x^{\frac{1}{4}}=\sqrt[4]{x}$$
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$$x^{\frac{1}{3}}=\sqrt[3]{x}$$
See Page on Variables and Functions