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Exponentiation
DEFINITION: If \latex{a} is an arbitrary real number, and m is a positive integer greater than 1, then the power \latex{a^{m}} means the product with m factors, where all factors are a.
If \latex{m} \latex{ = 1 }, then according to the definition: \latex{a^1=a}.
\latex{a^{m}= \underbrace{a\times a\times a\times \dots\times a}_{m\,\text{ factors}}}
index or power
or exponent
or exponent
base
product with \latex{ m } factors
For example:
Powers of 2
\latex{2^2=4}
\latex{2^3=8}
\latex{2^4=16}
\latex{2^6=64}
\latex{2^5=32}
\latex{2^7=128}
\latex{2^8=256}
\latex{2^9=512}
\latex{2^{10}=1,024}
\latex{2^{11}=2,048}
\latex{3^{5} = 3\times3\times 3\times3\times 3=243}
\latex{5^{3} = 5\times 5\times 5= 125}
\latex{\left(- 3\right)^{2} =\left(-3\right) × \left(-3\right)=9}
\latex{\left(- 3\right)^{3} =\left(-3\right) × \left(-3\right) × \left(-3\right)=-27}
Powers of \latex{3}
\latex{3^2=9}
\latex{3^3=27}
\latex{3^4=81}
\latex{3^5=243}
\latex{3^6=729}
\latex{3^7=2,187}
\latex{\left(\frac{2}{3}\right) ^{3} =\frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} =\frac{8}{27} }
\latex{\left(-\frac{3}{4}\right) ^{2} =\left(-\frac{3}{4} \right) \times \left(-\frac{3}{4} \right) =\frac{9}{16}}
Powers of \latex{5}
\latex{5^2=25}
\latex{5^3=125}
\latex{5^4=625}
\latex{5^5=3,125}
The exponential identities
IDENTITY I: The product of powers with the same base equals to the power where the common base is raised to the sum of the indices:
\latex{a^{m} \times a^{n} =a^{m+n}}, where \latex{a\in \R}; \latex{m,n\in \N^{+}}.
multiplication of powers with the same base
\latex{a^{m}\times a^{n}=a^{m+n}}
Proof
\latex{a^{m}\times a^{n}= \underbrace{\underbrace{\left(a\times a\times \dotsc\times a\right)\times }_{m\;\text{factors}}\underbrace{\left(a\times a\times \dotsc\times a\right) }_{n\;\text{factors}}}_{m+ n\;\text{factors}}=a^{m+n}}
definition
associativity
and definition
and definition
IDENTITY II: When dividing powers with the same base the fraction can be simplified, the result depends on whether the index of the numerator or the denominator is greater:
\latex{\frac{a^m}{a^n}=\begin{cases}a^{m-n}, \;\,\text{if}\;m\gt n,\\1, \;\,\,\,\,\,\,\,\,\,\,\,\,\,\text{if} \;m=n,\\\dfrac{1}{a^{n-m}}, \;\,\,\,\,\text{if}\; m\lt n,\end{cases}}
where \latex{a\in \R;a\neq 0} and \latex{m,n\in \N^{+}}.
division of powers with the same base
\latex{\frac{a^m}{a^n}=\begin{cases}a^{m-n}, \;\text{if}\;m\gt n\\1, \;\,\,\,\,\,\,\,\,\,\,\text{if} \;m=n\\\dfrac{1}{a^{n-m}}, \;\,\,\,\text{if}\; m\lt n\end{cases}}
Proof
\latex{\frac{a^m}{a^n}=\frac{\overbrace{a\times a\dots\times a}^{m \,\text{factors}}}{\underbrace {a\times a\dots\times a}_{n \, \text{factors}}}=\begin{cases}a^{m-n}, \;\text{if}\; m\gt n, \\1, \;\,\,\,\,\,\,\,\,\,\, \text{if}\, m=n,\\\dfrac{1}{a^{n-m}}, \;\,\,\,\text{if}\; m\lt n.\end{cases} }
definition
simplification
and definition
and definition
IDENTITY III: The power of a product is equal to the product of the powers of the factors:
\latex{\left(a\times b\right)^{n} = a^{n} \times b^{n}}, where \latex{a,b \in \R; n\in \N^{+}}.
exponentiation of a product
\latex{(a\times b)^n=a^n\times b^n}
Proof
\latex{\left(a\times b\right) ^{n} =\underbrace{\left(a\times b\right)\times \left(a\times b\right)\times \dotsc\times \left(a\times b\right)}_{n\;\text{factors}}=}
\latex{=\underbrace{\left(a\times a\times \dotsc\times a\right)}_{n\;\text{factors}}\times \underbrace{\left(b\times b\times \dotsc\times b\right)}_{n\;\text{factors}}=a^{n} \times b^{n}}.
\latex{=\underbrace{\left(a\times a\times \dotsc\times a\right)}_{n\;\text{factors}}\times \underbrace{\left(b\times b\times \dotsc\times b\right)}_{n\;\text{factors}}=a^{n} \times b^{n}}.
definition
commutativity,
associativity
associativity
definition
IDENTITY IV: The power of \latex{ a } fraction is equal to the quotient of the power of the numerator and of the denominator:
\latex{\left(\frac{a}{b}\right)^{n} = \frac{a^{n} }{b^{n} }}, where \latex{a,\,b\in \R; b\neq 0; n\in \N^{+}}.
exponentiation of a fraction
\latex{\left(\frac{a}{b}\right)^{n} = \frac{a^{n} }{b^{n} }}
Proof
\latex{\left(\frac{a}{b}\right)^n=\left(\frac{a}{b}\right)\times\left(\frac{a}{b}\right)\times \dots\times\left(\frac{a}{b}\right)=\frac{\overbrace{a\times a\times\dots\times a}^{n\, \text{factors}}}{\underbrace{b\times b\times\dots\times b}_{n\,\text{factors}}}=\frac{a^n}{b^n}}.
definition
definition
multiplication
of fractions
of fractions
IDENTITY V: The power of \latex{ a } power is equal to the power where the base is raised to the product of the indices:
\latex{\left(a^{n} \right)^{m} = a^{m\times n}}, if \latex{a\in \R; m,n\in \N^{+}}.
exponentiation of a power
\latex{\left(a^{n} \right)^{m} = a^{m\times n}}
Proof
\latex{(a^n)^m=\underbrace{(a^n)\times(a^n)\times\dots\times(a^n)}_{m\, \text{factors}}=}
\latex{=\underbrace{\overbrace{(a\times a \times \dots \times a)}^{n\, \text{factors}}\times \overbrace{(a\times a \times \dots \times a)}^{n\,\text{factors}} \times\dots\times\overbrace{(a\times a \times \dots \times a)}^{n\,\text{factors}} }_{m\times n\,\text{factors}}= a^{m\times n}}.
\latex{=\underbrace{\overbrace{(a\times a \times \dots \times a)}^{n\, \text{factors}}\times \overbrace{(a\times a \times \dots \times a)}^{n\,\text{factors}} \times\dots\times\overbrace{(a\times a \times \dots \times a)}^{n\,\text{factors}} }_{m\times n\,\text{factors}}= a^{m\times n}}.
definition
definition
associativity
and definition
and definition
When adding or subtracting powers we can reach our goal only by calculating the powers, and the result usually cannot be expressed as a power.
For example \latex{5^3 + 5^2 = 125 + 25 = 150}, and 150 cannot be expressed as a power of 5 with integer index.
Example 1
Let us calculate: \latex{\frac{7^{2}\times \left(5^{2}\times 7^{3}\right)^{5} }{\left(7^{2}\times 5^{3}\right)^{6} } \times \left(\frac{5^{3} }{7^{2}} \right)^{4}}.
Solution
\latex{\frac{7^{2}\times \left(5^{2}\times 7^{3}\right)^{5}}{\left(7^{2}\times 5^{3}\right)^{6}} \times\left(\frac{5^{3}}{7^{2} } \right)^{4} = \frac{7^{2}\times \left(5^{2}\right)^{5}\times \left(7^{3}\right)^{5}}{\left(7^{2}\right)^{6}\times \left(5^{3}\right)^{6} } \times \frac{\left(5^{3}\right)^{4} }{\left(7^{2}\right)^{4}} =}
\latex{=\frac{7^{2}\times 5^{10}\times 7^{15}\times 5^{12}}{7^{12}\times 5^{18}\times 7^{8}} =\frac{7^{17}\times 5^{22}}{7^{20}\times 5^{18}} =\frac{5^{4}}{7^{3}} =\frac{625}{343}}
Identity III and IV
Identity V,
multiplication
of fractions
multiplication
of fractions
Identity I
Identity II
Example 2
Let us calculate the value of \latex{(2^4+2^4)^5} and \latex{(3^4-3^2)^4}.
Solution
Since a sum cannot be raised to a power by terms, first the operation of addition should be done: \latex{2^4+2^4=2\times 2^4=2^5}. The result of the operation is \latex{\left(2^5\right)^5=2^{25}}.
A difference cannot be raised to a power by terms either:
A difference cannot be raised to a power by terms either:
\latex{\left(3^4-3^2\right)^4=\left[3^2\left(3^2-1\right)\right]^4=\left(3^2\times8\right)^4=\left(3^2\times2^3\right)^4=3^8\times2^{12}}.
In general: \latex{(a+b)^n\neq a^n+b^n}, as for example \latex{\left(2+3\right)^4=5^4=625}, but \latex{2^4+3^4=16+81=97.}
Example 3
Let us simplify expression \latex{\frac{\left(a^3\times b\right)^4\times\left(a^2\right)^3}{\left(b^2\right)^5\times\left(a^2b\right)^3}}, where \latex{a\neq0;\,b\neq0}.
Solution
\latex{\frac{(a^3b)^4\times (a^2)^3}{(b^2)^5\times(a^2b)^3}=\frac{(a^3)^4\times b^4\times(a^2)^3}{(b^2)^5\times (a^2)^3\times b^3}=}
Identity III
Identity V
\latex{=\frac{a^{12}\times b^4 \times a^6}{b^{10}\times a^6\times b^3}=\frac{a^{18}\times b^4}{a^6\times b^{13}}=\frac{a^{12}}{b^9}}
Identity I
Identity II
Example 4
Which number is greater: \latex{A = \left(6^3\right)^2\times5^6} or \latex{B=2^8\times15^6?}
Solution
Certainly our aim is not to calculate these products. Let us transform both expressions:
\latex{A=\left(2^3\times3^3\right)^2\times 5^6=2^6\times3^6\times5^6}
\latex{B=2^8\times\left(3\times5\right)^6=2^8\times3^6\times5^6=2^2\times A}
Thus \latex{B\gt A.}
Exercises
{{exercise_number}}. Decide which number is greater.
- \latex{5^{12}} or \latex{\left(5^5\right)^2}
- \latex{2^4\times2^5} or \latex{\left(2^4\right)^2}
- \latex{\left(\frac{2}{3}\right)^4} or \latex{\frac{16}{3^4}}
- \latex{\left(3^2\right)^3} or \latex{\left(3^2\times 3^3\right)^2}
- \latex{15^9} or \latex{9^{15}}
- \latex{125^4\times 64^3} or \latex{100^7}
{{exercise_number}}. Calculate the values of the following expressions.
- \latex{\left(2^3\times5\right)^3}
- \latex{\frac{\left(7^3\right)^5}{7^{12}}}
- \latex{\frac{\left(2^4\times5\right)^6\times2^3}{\left(5^2\times2^3\right)^3}\times\frac{\left(2\times5^3\right)^4}{\left(2^2\times5\right)^{12}}}
- \latex{\frac{\left(3\times7^3\right)^4}{3^6}\times\frac{9^5\times3^8}{49^6}}
- \latex{\left(\frac{16}{27}\right)^3\times \frac{\left(3^4\times2\right)^5}{\left(3^3\right)^4\times\left(2^5\right)^3}}
- \latex{\frac{32^3\times625^2}{128^4}\times\frac{64^5}{25^6}}
- \latex{\left(2\times3^2+5\right)^2}
- \latex{\frac{\left(3\times2^4+1\right)^4}{\left(7^3\right)^3}}
{{exercise_number}}. Simplify the following expressions for all possible values of the variables.
- \latex{\left(a^2b\right)^3}
- \latex{\frac{\left(a^4\right)^3}{a^7}}
- \latex{\frac{\left(a^3\times b\right)^3\times b^5}{\left(b^2\right)^3\times a^8}}
- \latex{\frac{\left(x^2y^3\right)^5\times \left(xy^2\right)^3}{\left(x^2\right)^2\times\left(yx^2\right)^4\times y^{15}}}
- \latex{\left(\frac{2x}{y}\right)^3\times \left(\frac{xy^2}{2}\right)^2\times\left(\frac{1}{x}\right)^4}
- \latex{\frac{\left(ab^2\right)^4\times\left(a^3\right)^2b}{\left(b^2a^3\right)^3}\times\frac{a^3b}{\left(b^3\right)^2}}
- \latex{\frac{\left(a^3b\right)^4\times b^3}{\left(ab^2\right)^5\times a^2}\div\frac{\left(a^5\right)^2b}{\left(a^4\times b^3\right)^2}}
{{exercise_number}}. How many digits are there in the below numbers written in the decimal number system?
- \latex{10^{2,000}-5\times10^{1,000}}
- \latex{\left(10^5\right)^7+\left(10^3\right)^8}
- \latex{2^{31}\times 5^{32}}
- \latex{2^{15}\times5^{15}-2^{13}\times 5^{12}}
Puzzle
Give numbers \latex{a, b, c} so that \latex{10^{a}+3\times 10^{b}+2\times 10^{c}= 32,100.}
