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Power and root
The power and root are important for modeling interest rates, rates of return, and growth. In this chapter, we provide the basics.
1. Power with positive integers in the exponentIf a real number a is mutiplied n-times with itself, then there is a shorthand term for this:
is called the power with basis a and exponent n.
Examples
\(2^3=2\cdot 2\cdot 2=8 \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad 0.1^4=0.1\cdot 0.1\cdot 0.1\cdot 0.1=0.0001\)
\((-3)^3=(-3)\cdot (-3)\cdot (-3)=-27 \quad \quad \quad (-3)^4=(-3)\cdot (-3)\cdot (-3)\cdot (-3)=81\)
1. Multiplication of powers with an identical basis
$ \quad {{a}^{n}}\cdot {{a}^{m}}\,\,=\,\,{{a}^{n+m}}$
2. Raising a power to a higher power$ \quad {{\left( {{a}^{n}} \right)}^{m}}\,\,=\,\,{{a}^{n\cdot m}}$
3. The power of a product$ \quad {{\left( a\cdot b \right)}^{n}}\,\,=\,\,{{a}^{n}}\cdot {{b}^{n}}$
4. Division of powers with an identical basis
$\quad $ 
$ \quad {{\left( \frac{a}{b} \right)}^{n}}\,\,=\,\,\frac{{{a}^{n}}}{{{b}^{n}}}$
The laws have to be proved.
As an example, the proof for the first law is as follows::
Examples
${{2}^{3}}\cdot {{2}^{6}}\,\,\,\,\overset{(1)}{\mathop =}\,\,\,{{2}^{3+6}}\,\,=\,\,{{2}^{9}} \quad \quad \quad {{\left( {{a}^{4}} \right)}^{5}}\,\,\,\,\overset{(2)}{\mathop =}\,\,\,{{a}^{4\cdot 5}}\,\,=\,\,{{a}^{20}}$
${{\left( 2\cdot 3 \right)}^{4}}\,\,\,\,\overset{(3)}{\mathop =}\,\,\,{{2}^{4}}\cdot {{3}^{4}} \quad \quad \quad \quad \quad \quad \frac{{{2}^{8}}}{{{2}^{3}}}\,\,\,\,\overset{(4)}{\mathop =}\,\,\,{{2}^{8-3}}\,\,=\,\,{{2}^{5}}$
$\frac{{{2}^{3}}}{{{2}^{8}}}\,\,\,\,\overset{(4)}{\mathop =}\,\,\,\frac{1}{{{2}^{8-3}}}\,\,=\,\,\frac{1}{{{2}^{5}}} \quad \quad \quad \quad \quad \frac{{{15}^{3}}}{{{5}^{3}}}\,\,\,\,\overset{(5)}{\mathop =}\,\,\,{{\left( \frac{15}{5} \right)}^{3}}\,\,=\,\,{{3}^{3}}$
2. Power with an integer in the exponent
So far, we have admitted only natural numbers in the exponent. This is why we needed a case distinction for the fourth law. Once we admit any integer in the exponent, things will turn out to be less complicated.
The fourth law provides a hint for a reasonable definition:
$\frac{{{2}^{3}}}{{{2}^{5}}}\,\,=\,\,\frac{1}{{{2}^{2}}} \quad \quad or \quad \quad \frac{{{2}^{3}}}{{{2}^{5}}}\,\,=\,\,{{2}^{3-5}}\,\,=\,\,{{2}^{-2}}$
${{a}^{-n}}\,\,=\,\,\frac{1}{{{a}^{n}}}\,\,=\,\,{{\left( \frac{1}{a} \right)}^{n}} \quad \quad \quad a\in \mathbb{R}\backslash \left\{ 0 \right\},\,\,n\in \mathbb{N}$
${{10}^{-3}}\,\,=\,\,\frac{1}{{{10}^{3}}}\,\,=\,\,{{\left( \frac{1}{10} \right)}^{3}}\,\,=\,\,0.001 \quad \quad \quad {{0.25}^{-2}}\,\,=\,\,{{\left( \frac{1}{4} \right)}^{-2}}\,\,=\,\,{{\left( \frac{4}{1} \right)}^{2}}\,\,=\,\,16$
${{\left( \frac{1}{{{a}^{3}}} \right)}^{-5}}\,\,=\,\,{{\left( \frac{{{a}^{3}}}{1} \right)}^{5}}\,\,=\,\,{{a}^{15}} \quad \quad \quad {{\left( \frac{{{a}^{2}}}{{{b}^{3}}} \right)}^{-4}}\,\,=\,\,{{\left( \frac{{{b}^{3}}}{{{a}^{2}}} \right)}^{4}}\,\,=\,\,\frac{{{b}^{12}}}{{{a}^{8}}}$
So far, a suitable definition is missing for a power with an exponent of 0. We therefore present the following definition:
${{a}^{0}}\,\,=\,\,1 \quad \quad \quad a\in \mathbb{R}\backslash \left\{ 0 \right\}$
The five laws are also valid for a power with integers.
$\frac{{{a}^{n}}}{{{a}^{m}}}\,\,=\,\,{{a}^{n-m}} \quad \quad \quad m,n\in \mathbb{Z}$
${{a}^{-m}}\cdot {{a}^{-n}}\,\,\,\,\overset{(1)}{\mathop =}\,\,\,{{a}^{-n-m}}\quad \quad \quad {{\left( {{a}^{-n}} \right)}^{m}}\,\,\,\,\overset{(2)}{\mathop =}\,\,\,{{a}^{-mn}}$
$\frac{{{a}^{-n}}}{{{a}^{-m}}}\,\,\,\,\overset{(4)}{\mathop =}\,\,\,{{a}^{-n+m}} \quad \quad \quad \frac{{{a}^{-n}}}{{{a}^{m}}}\,\,\,\,\overset{(4)}{\mathop =}\,\,\,{{a}^{-n-m}}$
3. Formula for a compound interest
A certain capital stock is invested for several years with a compound interest.
Given: starting capital stock (present value) \({K_0}\), interest rate \(i\), maturity \(n\) years
Wanted: future value \({K_n}\)
We calculate the future value after ...
... one year: ${{K}_{1}}\,\,=\,\,{{K}_{0}}+{{K}_{0}}\cdot i\,\,=\,\,{{K}_{0}}\cdot (1+i)$ while \((1+i)\) is the interest factor
... two years:: ${{K}_{2}}\,\,=\,\,{{K}_{1}}+{{K}_{1}}\cdot i\,\,=\,\,{{K}_{1}}\cdot (1+i)\,\,=\,\,{{K}_{0}}\cdot {{(1+i)}^{2}}$
... three years: ${{K}_{3}}\,\,=\,\,{{K}_{2}}+{{K}_{2}}\cdot i\,\,=\,\,{{K}_{2}}\cdot (1+i)\,\,=\,\,{{K}_{0}}\cdot {{(1+i)}^{3}}$
Generally, after n years:
${{K}_{n}}\,\,=\,\,{{K}_{0}}\cdot {{(1+i)}^{n}}$
Example
For \({K_0}=2000 \quad i=0.05 \quad n= 10\), we get:
${{K}_{10}}\,\,=\,\,2000\cdot {{1.05}^{10}}\,\,=\,\,3257.79$
4. Power with a fraction in the exponent and roots
Introductory examples
$1. \quad {{K}_{0}}\,\,=\,\,1000.- \quad {{K}_{2}}\,\,=\,\,1060.90 \quad i=?$
$2. \quad {{K}_{0}}\,\,=\,\,1000.- \quad {{K}_{8}}\,\,=\,\,1368.70 \quad i=?$
In these examples, we are looking for the number \(1+i\), which - multiplied by itself twice and eight times respectively - results in \(1.0609=({{K}_{2}}/ {{K}_{0}})\) and \(1.3687=({{K}_{8}}/ {{K}_{0}})\) respectively. Such numbers are named roots.
We write:
According to the exercise we have to calculate the interest rate \(i\) and not the interest factor \(1+i\). Therefore,
\(i=0.03=3\% \quad \) and \( \quad i=0.04=4\%\).
The n'th root of a positive number a is a positive number z whose n'th power equals a.
$z\,\,=\,\,\sqrt[n]{a} \quad if \quad {{z}^{n}}\,\,=\,\,a$
In $z\,\,=\,\,\sqrt[n]{a} \quad $, a is called radicand, n denotes the order of the root and z is the root.
The operation is called root extraction.
Remarks
• The definition of a root does not show how actually deremine it. It only shows the characteristics a root must have.
• The n'th root of a negative number is not defined.
• For $\sqrt[2]{a} $, the order of the root is neglected. One writes $\sqrt{a}$.
• The root extraction with n as the order of the root is the inverse operation of raising to the power of n: ${{\left( \sqrt[n]{a} \right)}^{n}}\,\,=\,\,a\,\,=\,\,\sqrt[n]{{{a}^{n}}}$
Examples
How can roots be calculated with the calculator or Excel?
For this purpose the following considerations might be helpful:
We write $\sqrt[n]{a}$ as a power with a (preliminarily) unknown exponent:
$\sqrt[n]{a}\,\,=\,\,{{a}^{x}}$According to the fourth remark:
${{\left( \sqrt[n]{a} \right)}^{n}}\,\,=\,\,{{\left( {{a}^{x}} \right)}^{n}}\,\,=\,\,{{a}^{xn}}\,\,=\,\,a\,\,={{a}^{1}}$Therefore, $x\cdot n=1$ and $x\,\,=\,\,\frac{1}{n}$
Thus, we get the following definition:
For the second introductory example, we get:
Exercises for section 1. Power with positive integers in the exponent
1)
$ \large \frac{{{a}^{3}}\cdot {{a}^{7}}}{{{a}^{21}}}\,\,=$
2)
${{\left( {{x}^{3}} \right)}^{4}}\cdot {{x}^{5}}\,\,=$
3)
$ \large \frac{{{\left( 2a{{b}^{3}} \right)}^{4}}}{{{\left( 3{{a}^{2}}b \right)}^{2}}}\,\,=$
4)
$ \large \frac{{{a}^{n+2}}}{{{a}^{n}}}\,\,=$
5)
$ \large \frac{{{3}^{n+2}}}{{{3}^{n+3}}}=$
Exercises for section 2. Power with an integer in the exponent
1)
$ {{2}^{-3}}\,\,=$
2)
$ {{10}^{-4}}\,\,=$
3)
${{7}^{-3}}\cdot {{7}^{2}}\,\,=$
4)
$ {{6}^{-2}}\cdot {{6}^{-1}}\,\,=$
5)
$\large \frac{{{7}^{3}}}{{{7}^{-3}}}\,\,=$
6)
$ \large \frac{{{x}^{-3}}}{{{x}^{7}}}\,\,=$
7)
$\large {{\left( \frac{2{{a}^{2}}}{{{(2b)}^{3}}} \right)}^{-3}}\,\,=$
8)
$ \large {{\left( \frac{{{x}^{-3}}}{{{y}^{2}}} \right)}^{-5}}\,\,=$
9)
$ 1.2\cdot {{10}^{-6}}=$
10)
$ 1.2\cdot {{10}^{6}}=$
Exercises for section 4. Power with a fraction in the exponent and roots
1)
$ \sqrt[5]{32}\,\,=$
2)
3)
For which interest rate does the capital double in 10 years?
4)
A firm doubles its revenues within 5 years.
a)
Calculate the average yearly increase in revenues (in %).
b)
How large is the percentaged increase per month?
Two useful websites covering the theory of power and roots can be found on: