In general, if \(a\) is real number, and \(m\) and \(n\) are positive integers, then Law 1 of Indices: (same base) \({{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\) \({{6}^{4}}\times {{6}^{5}}= \bbox[border: 0.5px dotted #A1AFC9, 0.5px]{\phantom{0}}^{ \text{}\hspace{0.5em} \bbox[border: 0.5px dotted #A1AFC9, 0.5px]{\phantom{0}}}\) \({{a}^{3}}\times {{a}^{4}}= \bbox[border: 0.5px dotted #A1AFC9, 0.5px]{\phantom{0}}^{ \text{}\hspace{0.5em} \bbox[border: 0.5px dotted #A1AFC9, 0.5px]{\phantom{0}}}\) Law 2 of Indices: (same base) \({{a}^{m}}\div {{a}^{n}}={{a}^{m-n}}\), if \(a\ne 0\) \(\frac{{{10}^{6}}}{{{10}^{4}}}= \bbox[border: 0.5px dotted #A1AFC9, 0.5px]{\phantom{0}}^{ \text{}\hspace{0.5em} \bbox[border: 0.5px dotted #A1AFC9, 0.5px]{\phantom{0}}}\) \({{a}^{7}}\div {{a}^{3}}= \bbox[border: 0.5px dotted #A1AFC9, 0.5px]{\phantom{0}}^{ \text{}\hspace{0.5em} \bbox[border: 0.5px dotted #A1AFC9, 0.5px]{\phantom{0}}}\) Law 3 of Indices: (same base) \({{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}\) \({{\left( {{2}^{5}} \right)}^{2}} = \bbox[border: 0.5px dotted #A1AFC9, 0.5px]{\phantom{0}}^{ \text{}\hspace{0.5em} \bbox[border: 0.5px dotted #A1AFC9, 0.5px]{\phantom{0}}}\) \({{\left( {{a}^{3}} \right)}^{4}} = \bbox[border: 0.5px dotted #A1AFC9, 0.5px]{\phantom{0}}^{ \text{}\hspace{0.5em} \bbox[border: 0.5px dotted #A1AFC9, 0.5px]{\phantom{0}}}\) In general, if \(a\) and \(b\) are real numbers, and \(n\) is a positive integer, then Law 4 of Indices: (same index) \({{a}^{n}}\times {{b}^{n}}={{\left( a\times b \right)}^{n}}\) It can also be written as \({{\left( ab \right)}^{n}}={{a}^{n}}{{b}^{n}}\). \({{\left( -3 \right)}^{2}}\times {{\left( -4 \right)}^{2}= \left( { \bbox[border: 1px dotted #A1AFC9, 1px]{\phantom{0}}}\times{ \bbox[border: 1px dotted #A1AFC9, 1px]{\phantom{0}}} \right) ^{2}}\) Law 5 of Indices: (same index) \({{a}^{n}}\div {{b}^{n}}={{\left( \frac{a}{b} \right)}^{n}}\), if \(b\ne 0\) It can also be written as \({{\left( \frac{a}{b} \right)}^{n}}=\left( \frac{{{a}^{n}}}{{{b}^{n}}} \right)\). \({{8}^{3}}\div {{5}^{3}}= \left( \frac{ \bbox[border: 1px dotted #A1AFC9, 1px]{\phantom{0}}}{ \bbox[border: 1px dotted #A1AFC9, 1px]{\phantom{0}}} \right)^{3}\)

Laws of Indices

Secondary 3

In general, if \(a\) is real number, and \(m\) and \(n\) are positive integers, then

Law 1 of Indices: (same base)	\({{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\)	\({{6}^{4}}\times {{6}^{5}}= \bbox[border: 0.5px dotted rgb(161 175 201), 0.5px]{\phantom{0}}^{ \text{}\hspace{0.5em} \bbox[border: 0.5px dotted rgb(161 175 201), 0.5px]{\phantom{0}}}\) \({{a}^{3}}\times {{a}^{4}}= \bbox[border: 0.5px dotted rgb(161 175 201), 0.5px]{\phantom{0}}^{ \text{}\hspace{0.5em} \bbox[border: 0.5px dotted rgb(161 175 201), 0.5px]{\phantom{0}}}\)
Law 2 of Indices: (same base)	\({{a}^{m}}\div {{a}^{n}}={{a}^{m-n}}\), if \(a\ne 0\)	\(\frac{{{10}^{6}}}{{{10}^{4}}}= \bbox[border: 0.5px dotted rgb(161 175 201), 0.5px]{\phantom{0}}^{ \text{}\hspace{0.5em} \bbox[border: 0.5px dotted rgb(161 175 201), 0.5px]{\phantom{0}}}\) \({{a}^{7}}\div {{a}^{3}}= \bbox[border: 0.5px dotted rgb(161 175 201), 0.5px]{\phantom{0}}^{ \text{}\hspace{0.5em} \bbox[border: 0.5px dotted rgb(161 175 201), 0.5px]{\phantom{0}}}\)
Law 3 of Indices: (same base)	\({{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}\)	\({{\left( {{2}^{5}} \right)}^{2}} = \bbox[border: 0.5px dotted rgb(161 175 201), 0.5px]{\phantom{0}}^{ \text{}\hspace{0.5em} \bbox[border: 0.5px dotted rgb(161 175 201), 0.5px]{\phantom{0}}}\) \({{\left( {{a}^{3}} \right)}^{4}} = \bbox[border: 0.5px dotted rgb(161 175 201), 0.5px]{\phantom{0}}^{ \text{}\hspace{0.5em} \bbox[border: 0.5px dotted rgb(161 175 201), 0.5px]{\phantom{0}}}\)

Law 1 of Indices:

(same base)

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

\({{6}^{4}}\times {{6}^{5}}= \bbox[border: 0.5px dotted rgb(161 175 201), 0.5px]{\phantom{0}}^{ \text{}\hspace{0.5em} \bbox[border: 0.5px dotted rgb(161 175 201), 0.5px]{\phantom{0}}}\)
\({{a}^{3}}\times {{a}^{4}}= \bbox[border: 0.5px dotted rgb(161 175 201), 0.5px]{\phantom{0}}^{ \text{}\hspace{0.5em} \bbox[border: 0.5px dotted rgb(161 175 201), 0.5px]{\phantom{0}}}\)

Law 2 of Indices:

(same base)

\({{a}^{m}}\div {{a}^{n}}={{a}^{m-n}}\), if \(a\ne 0\)

\(\frac{{{10}^{6}}}{{{10}^{4}}}= \bbox[border: 0.5px dotted rgb(161 175 201), 0.5px]{\phantom{0}}^{ \text{}\hspace{0.5em} \bbox[border: 0.5px dotted rgb(161 175 201), 0.5px]{\phantom{0}}}\)
\({{a}^{7}}\div {{a}^{3}}= \bbox[border: 0.5px dotted rgb(161 175 201), 0.5px]{\phantom{0}}^{ \text{}\hspace{0.5em} \bbox[border: 0.5px dotted rgb(161 175 201), 0.5px]{\phantom{0}}}\)

Law 3 of Indices:

(same base)

\({{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}\)

\({{\left( {{2}^{5}} \right)}^{2}} = \bbox[border: 0.5px dotted rgb(161 175 201), 0.5px]{\phantom{0}}^{ \text{}\hspace{0.5em} \bbox[border: 0.5px dotted rgb(161 175 201), 0.5px]{\phantom{0}}}\)
\({{\left( {{a}^{3}} \right)}^{4}} = \bbox[border: 0.5px dotted rgb(161 175 201), 0.5px]{\phantom{0}}^{ \text{}\hspace{0.5em} \bbox[border: 0.5px dotted rgb(161 175 201), 0.5px]{\phantom{0}}}\)

In general, if \(a\) and \(b\) are real numbers, and \(n\) is a positive integer, then

Law 4 of Indices: (same index)	\({{a}^{n}}\times {{b}^{n}}={{\left( a\times b \right)}^{n}}\) It can also be written as \({{\left( ab \right)}^{n}}={{a}^{n}}{{b}^{n}}\).	\({{\left( -3 \right)}^{2}}\times {{\left( -4 \right)}^{2}= \left( { \bbox[border: 1px dotted rgb(161 175 201), 1px]{\phantom{0}}}\times{ \bbox[border: 1px dotted rgb(161 175 201), 1px]{\phantom{0}}} \right) ^{2}}\)
Law 5 of Indices: (same index)	\({{a}^{n}}\div {{b}^{n}}={{\left( \frac{a}{b} \right)}^{n}}\), if \(b\ne 0\) It can also be written as \({{\left( \frac{a}{b} \right)}^{n}}=\left( \frac{{{a}^{n}}}{{{b}^{n}}} \right)\).	\({{8}^{3}}\div {{5}^{3}}= \left( \frac{ \bbox[border: 1px dotted rgb(161 175 201), 1px]{\phantom{0}}}{ \bbox[border: 1px dotted rgb(161 175 201), 1px]{\phantom{0}}} \right)^{3}\)

Law 4 of Indices:

(same index)

\({{a}^{n}}\times {{b}^{n}}={{\left( a\times b \right)}^{n}}\)

It can also be written as \({{\left( ab \right)}^{n}}={{a}^{n}}{{b}^{n}}\).

\({{\left( -3 \right)}^{2}}\times {{\left( -4 \right)}^{2}= \left( { \bbox[border: 1px dotted rgb(161 175 201), 1px]{\phantom{0}}}\times{ \bbox[border: 1px dotted rgb(161 175 201), 1px]{\phantom{0}}} \right) ^{2}}\)

Law 5 of Indices:

(same index)

\({{a}^{n}}\div {{b}^{n}}={{\left( \frac{a}{b} \right)}^{n}}\), if \(b\ne 0\)

It can also be written as \({{\left( \frac{a}{b} \right)}^{n}}=\left( \frac{{{a}^{n}}}{{{b}^{n}}} \right)\).

\({{8}^{3}}\div {{5}^{3}}= \left( \frac{ \bbox[border: 1px dotted rgb(161 175 201), 1px]{\phantom{0}}}{ \bbox[border: 1px dotted rgb(161 175 201), 1px]{\phantom{0}}} \right)^{3}\)

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