Topic 6 Indices

6.1 Exponents

Subscripts and superscripts are also called indices (singular index). Subscripts are usually used to denote families of variables, for example \(a_1,a_2,a_3,\dots\) (instead of \(a,b,c\)).

A superscript usually denotes an exponent, i.e. it represents the power to which a given number is raised.

Examples:

  • \(4^3=\underbrace{4\cdot4\cdot4}_{3\text{ times}}=64\)
  • \(2^5=\underbrace{2\cdot2\cdot2\cdot2\cdot2}_{5\text{ times}}=32\)
  • \(x^4=\underbrace{x\cdot x\cdot x\cdot x}_{4\text{ times}}\)

In general, we define:

  • \(x^n := \underbrace{x\cdot x\cdots x}_{n\text{ times}}\), when \(n>0\),
  • \(x^0 := 1\),
  • \(x^{-n} := \dfrac{1}{\underbrace{x\cdot x\cdots x}_{n\text{ times}}}\), when \(n>0\).

Further examples:

  • \(x^{\frac{1}{2}} := \sqrt{x}\)   for \(x\geq 0\),
  • \(x^{\frac{1}{3}} := \sqrt[3]{x}\).

In general, we define:

  • \(x^{\frac{1}{n}} := \sqrt[n]{x}\)    for \(x\geq0\) and \(n>0\),
  • \(x^{\frac{m}{n}} := \sqrt[n]{x^m}\)    for \(x\geq0\) and \(m,n>0\),
  • \(x^{-\frac{m}{n}} := \dfrac{1}{\sqrt[n]{x^m}}\)    for \(x\geq0\) and \(m,n>0\).

6.2 Laws of exponents

The basic ones are:

  1. \(a^r\cdot a^s = a^{r+s}\)
  2. \((a\cdot b)^r = a^{r}\cdot b^{r}\)
  3. \(\left(a^r\right)^s = a^{r\cdot s}\)

From these, one can derive the following two:

  1. \(\frac{a^r}{a^s} = a^{r-s}\)
  2. \(\left(\sqrt[n]{a}\right)^m = a^{\frac{m}{n}}\)

Deriving 4: \(\frac{a^r}{a^s} = a^r\cdot \frac{1}{a^s} = a^r\cdot a^{-s} \stackrel{1.}{=} a^{r-s}\).

Deriving 5: \(\left(\sqrt[n]{a}\right)^m = \left(a^{\frac{1}{n}}\right)^m \stackrel{3.}{=} a^{\frac{1}{n}\cdot m} = a^\frac{m}{n}\).

Example 6.1 Write the following expressions using only positive indices: \(x^{-4}\),  \(3x^{-4}\),  \((3x)^{-4}\)   and \(\frac{2}{y^{-3}}\).

Solution. Calculate

  • \(x^{-4}=\frac{1}{x^4}\)
  • \(3x^{-4}=3\cdot\frac{1}{x^4}=\frac{3}{x^4}\)
  • \((3x)^{-4} = \frac{1}{(3x)^4} = \frac{1}{3^4\cdot x^4}=\frac{1}{81x^4}\)
  • \(\frac{2}{y^{-3}}=\frac{2}{\frac{1}{y^3}}=2y^3\)

Example 6.2 Write the following expressions using only single index: \(\left(x^2\right)^{-3}\),   \(\left((-x)^{-2}\right)^{-3}\),   \(\left(\frac{t}{t^{-2}}\right)^{4}\).

Solution. Calculate

  • \(\left(x^2\right)^{-3} = x^{2\cdot(-3)}=x^{-6}\)
  • \(\left((-x)^{-2}\right)^{-3} = (-x)^{(-2)\cdot(-3)}=(-x)^{6}=\left((-1)\cdot x\right)^6=(-1)^6\cdot x^6=x^6\)
  • \(\left(\frac{t}{t^{-2}}\right)^{4}=\frac{t^4}{(t^{-2})^4}=\frac{t^4}{t^{-8}}=t^{4-(-8)}=t^{12}\)

Example 6.3 Simplify the expression \(A=\dfrac{x^{-6}y^{3/2}w^2}{x^{-3}w^3}\div\dfrac{x^\frac12\sqrt[3]{y}}{(xy)^3(\sqrt[4]{w})^{-2}}\) so that it uses only positive indices.

Solution. \[\begin{align*} A&= \frac{x^{-6}y^{\frac{3}{2}}w^2\,(xy)^3(\sqrt[4]{w})^{-2}}{x^{-3}w^3\, x^{\frac{1}{2}}\sqrt[3]{y}}\\ &=\frac{x^{-6}y^{\frac{3}{2}}w^2x^3y^3w^{-\frac12}}{x^{-3}w^3x^{\frac12}y^{\frac13}}\\ &=x^{-6+3-(-3)-\frac12}\,y^{\frac32+3-\frac13}\, w^{2-\frac12-3}\\ &=x^{-\frac12}\,y^{\frac{25}{6}}\,w^{-\frac32}\\ &=\frac{y^{\frac{25}{6}}}{x^{\frac12}\,w^{\frac32}}. \end{align*}\]

6.3 Scientific notation of numbers

Standard (or normalised) scientific notation of number is:

\(a\cdot 10^n\),     where \(1\leq|a|<10\) and \(n\) is an integer

Engineering scientific notation is:

\(a\cdot 1000^n\),     where \(1\leq|a|<1000\) and \(n\) is a multiple of \(3\)

The “\(a\)” is called mantissa, the “\(n\)” is called exponent.

Decimal expansion Standard notation Engineering notation
\(17\) \(1.7\cdot 10^1\) \(17\)
\(620\) \(6.2\cdot 10^2\) \(620\)
\(342567\) \(3.42567\cdot 10^5\) \(342.567\cdot 10^3\)
\(0.0001\) \(1\cdot 10^{-4}\) \(100\cdot 10^{-6}\)
\(0.00005\) \(5\cdot 10^{-5}\) \(50\cdot 10^{-6}\)