| |
| 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) |
|
\(\begin{aligned}4\,096^{\frac{5}{6}}&={\sqrt[6]{4\,096^5}} \\\\&=(\sqrt[6]{4\,096})^5. \end{aligned}\) |
|
|
