## Law of Indices

1.2  Law of Indices

Multiplication:

• In general, $$a^m \times a^n = a^{m+n}$$.

 Example (i) \begin{aligned} 2^3\times 2^2 &= 2^{3+2} \\\\&= 2^5. \end{aligned} (ii) \begin{aligned} &\space2^3 \times 6^3 \times 2^5 \times 6^2 \\\\&=2^3 \times 2^5 \times 6^3 \times 6^2 \\\\&= 2^{3+5} \times 6^{3+2} \\\\&= 2^8 \times 6^5. \end{aligned}

Division:

• In general, $$a^m \div a^n = a^{m-n}$$.

 Example (i) \begin{aligned}5^4 \div 5^2&= 5^{4-2} \\\\&=5^2. \end{aligned} (ii) \begin{aligned} &\space 20 k^5 y^3 \div 5 k^2 y \\\\&= \dfrac{20}{5} k^{5-2} y^{3-1} \\\\&=4 k^3 y^2. \end{aligned}

Power:

• In general, $$(a^m)^n = a ^{mn}$$.

 Example \begin{aligned} (8^5)^3&= 8^{5(3)} \\\\&= 8^{15}. \end{aligned}

Using law of indices to perform operations of multiplication and division:

• \begin{aligned} (a^m\times b^n)^q&=(a^m)^q\times (b^n)^q \\\\&=a^{mq}\times b^{nq} \\\\&=a^{mq}\,b^{nq} \end{aligned}

• \begin{aligned} (a^m\div b^n)^q&=(a^m)^q\div (b^n)^q \\\\&=a^{mq}\div b^{nq} \\\\&=\dfrac{a^{mq}}{b^{nq}} \end{aligned}

 Example (i) \begin{aligned} \bigg(\dfrac {3^4}{5^6}\bigg)^2 &=\dfrac {3^{4(2)}} {5^{6(2)}} \\\\&=\dfrac {3^8} {5^{12}}. \end{aligned} (ii) \begin{aligned} &\space (3 m^4 n^5)^3 \\\\&= (3 ^3) m^{4(3)} n^{5(3)} \\\\&= 27 m^{12} n^{15} \end{aligned}

Zero index:

• A number or an algebraic term with a zero index will give a value of $$1$$.
• In general, $$a^0=1\,\,;\,\,a\neq 0$$.

 Example (i) $$k^0 = 1$$ (ii) $$12^0 = 1$$

Negative index:

• A number or an algebraic term that has an index of a negative value.
• In general, $$a^{-n}=\dfrac{1}{a^n}\,\,;\,\,a\neq 0$$.

 Example (i) $$m^{-4} = \dfrac {1}{m^4}$$ (ii) $$4^{-2} = \dfrac {1}{4^2}$$ (iii) \begin{aligned} a^{-6}&= \dfrac {1}{a^6} \end{aligned} (iv) $$2 a^{-3} = \dfrac{2}{a^3}$$ (v) $$\dfrac{1}{4^3}= 4^{-3}$$

Fractional index:

• In general, $$\sqrt[n]{a}=a^{\frac{1}{n}}\,\,;\,\,a\neq 0$$.

 Example (i) $$\sqrt[5] {25} = 25 ^\frac{1}{5}$$ (ii) $$50^\frac{1}{5}= \sqrt[5]{50}$$

• $$a^{\frac{m}{n}} = (a^m)^\frac{1}{n}=(a^\frac{1}{n})^m$$

• $$a^{\frac{m}{n}} = {\sqrt[n] {a^m}}=({\sqrt [n]a)}^m$$

 Example (i) \begin{aligned}81^{\frac{3}{2}} &= (81^3)^{\frac{1}{2}} \\\\&=(81^{\frac{1}{2}})^3. \end{aligned} (ii) \begin{aligned}4\,096^{\frac{5}{6}}&={\sqrt[6]{4\,096^5}} \\\\&=(\sqrt[6]{4\,096})^5. \end{aligned}

## Law of Indices

1.2  Law of Indices

Multiplication:

• In general, $$a^m \times a^n = a^{m+n}$$.

 Example (i) \begin{aligned} 2^3\times 2^2 &= 2^{3+2} \\\\&= 2^5. \end{aligned} (ii) \begin{aligned} &\space2^3 \times 6^3 \times 2^5 \times 6^2 \\\\&=2^3 \times 2^5 \times 6^3 \times 6^2 \\\\&= 2^{3+5} \times 6^{3+2} \\\\&= 2^8 \times 6^5. \end{aligned}

Division:

• In general, $$a^m \div a^n = a^{m-n}$$.

 Example (i) \begin{aligned}5^4 \div 5^2&= 5^{4-2} \\\\&=5^2. \end{aligned} (ii) \begin{aligned} &\space 20 k^5 y^3 \div 5 k^2 y \\\\&= \dfrac{20}{5} k^{5-2} y^{3-1} \\\\&=4 k^3 y^2. \end{aligned}

Power:

• In general, $$(a^m)^n = a ^{mn}$$.

 Example \begin{aligned} (8^5)^3&= 8^{5(3)} \\\\&= 8^{15}. \end{aligned}

Using law of indices to perform operations of multiplication and division:

• \begin{aligned} (a^m\times b^n)^q&=(a^m)^q\times (b^n)^q \\\\&=a^{mq}\times b^{nq} \\\\&=a^{mq}\,b^{nq} \end{aligned}

• \begin{aligned} (a^m\div b^n)^q&=(a^m)^q\div (b^n)^q \\\\&=a^{mq}\div b^{nq} \\\\&=\dfrac{a^{mq}}{b^{nq}} \end{aligned}

 Example (i) \begin{aligned} \bigg(\dfrac {3^4}{5^6}\bigg)^2 &=\dfrac {3^{4(2)}} {5^{6(2)}} \\\\&=\dfrac {3^8} {5^{12}}. \end{aligned} (ii) \begin{aligned} &\space (3 m^4 n^5)^3 \\\\&= (3 ^3) m^{4(3)} n^{5(3)} \\\\&= 27 m^{12} n^{15} \end{aligned}

Zero index:

• A number or an algebraic term with a zero index will give a value of $$1$$.
• In general, $$a^0=1\,\,;\,\,a\neq 0$$.

 Example (i) $$k^0 = 1$$ (ii) $$12^0 = 1$$

Negative index:

• A number or an algebraic term that has an index of a negative value.
• In general, $$a^{-n}=\dfrac{1}{a^n}\,\,;\,\,a\neq 0$$.

 Example (i) $$m^{-4} = \dfrac {1}{m^4}$$ (ii) $$4^{-2} = \dfrac {1}{4^2}$$ (iii) \begin{aligned} a^{-6}&= \dfrac {1}{a^6} \end{aligned} (iv) $$2 a^{-3} = \dfrac{2}{a^3}$$ (v) $$\dfrac{1}{4^3}= 4^{-3}$$

Fractional index:

• In general, $$\sqrt[n]{a}=a^{\frac{1}{n}}\,\,;\,\,a\neq 0$$.

 Example (i) $$\sqrt[5] {25} = 25 ^\frac{1}{5}$$ (ii) $$50^\frac{1}{5}= \sqrt[5]{50}$$

• $$a^{\frac{m}{n}} = (a^m)^\frac{1}{n}=(a^\frac{1}{n})^m$$

• $$a^{\frac{m}{n}} = {\sqrt[n] {a^m}}=({\sqrt [n]a)}^m$$

 Example (i) \begin{aligned}81^{\frac{3}{2}} &= (81^3)^{\frac{1}{2}} \\\\&=(81^{\frac{1}{2}})^3. \end{aligned} (ii) \begin{aligned}4\,096^{\frac{5}{6}}&={\sqrt[6]{4\,096^5}} \\\\&=(\sqrt[6]{4\,096})^5. \end{aligned}