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As seen from their definitions, exponents and radicals are two closely related concepts which have more or less the same properties. It is very important to know and to master the properties of the exponents to succeed in solving the equations found in financial mathematics.
In the following definitions, it is important to keep in mind that
||\{a,b\} \subset \mathbb{R}\quad \text{and}\quad \{m,n\} \subset\mathbb{N}.||
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Definitions |
Examples |
|---|---|
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A positive integer exponent indicates the number of times the base appears in a multiplication. |
|2^{3}=2\times2\times2=8| |\left(\dfrac{1}{2}\right)^{4}=\dfrac{1}{2}\times \dfrac{1}{2}\times \dfrac{1}{2}\times \dfrac{1}{2}=\dfrac{1}{16}| |
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Any base raised to the exponent |0| equals |1| (except if the base is |0|). |
|4^{0}=1| |0^{0}\ \text{is undefined}| |
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Any base raised to an exponent of |1| equals the base itself. |
|25^{1}=25| |\left(\dfrac{8}{3} \right)^{1}=\dfrac{8}{3}| |
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Any base raised to a negative exponent is equivalent to the inverse of the base raised to a positive exponent. |
|2^{-4}=\dfrac{1}{2^{4}}| |\left(\dfrac{2}{3}\right)^{-5}=\left(\dfrac{3}{2}\right)^{5}| |
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A base assigned a fraction exponent results in a radical root. |
|8^{\dfrac{3}{5}}=\sqrt [5]{8^{3}}| |2^{\dfrac{1}{3}}=\sqrt [3]{2}| |
For the following properties, it is important to consider that
||\{a,b\} \subset \mathbb{R}\quad \text{and}\quad \{m,n\} \subset \mathbb{N}.||
In the following table, first the properties are defined and then examples are given. It is important to remember that an equality can be read from left to right as well as from right to left.
Transforming equalities in either direction is emphasized in the sections ahead.
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Properties |
Examples |
|---|---|
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If two powers with the same base are equal, then the exponents are equal. |
|8^{4}=8^{x}| therefore, |4=x.| |
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Multiplying powers with the same base: |
|8^{3}\times 8^{5}\times 8^{-2}=8^{3+5+^-2}=8^{6}| |
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Dividing powers with the same base: |
|\dfrac{4^{5}}{4^{3}}=4^{5-3}=4^{2}| |
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Power of a product: |
|(2xy)^{3}=2^{3}x^{3}y^{3}| |
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Power of a quotient: |
|\left(\dfrac{2}{3}\right)^{5}=\dfrac{2^{5}}{3^{5}}| |
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Power of a power: |
|(2^{3})^{3}=2^{3\times 3}=2^{9}| |
These properties are also applicable for certain other number sets.
If |a \in \mathbb{R}_+| and |\{m,n\} \subset \mathbb{Q}|, then |(a^m)^{^{\Large{n}}} = a^{m\times n}.|
||\begin{align} \left(3^{\dfrac{1}{2}}\right)^{\dfrac{3}{5}} &= 3^{^{\dfrac{1}{2} \times \dfrac{3}{5}}} \\ &= 3^{^\dfrac{3}{10}}\end{align}||
When multiplying, pay close attention to the value of the exponent. If the exponents are the same, the multiplication can be simplified. Otherwise, the two terms cannot be multiplied.
Possible to calculate the product (identical exponents)
||\begin{align} 4^\color{blue}{5} \times 3^\color{blue}{5} &= (4\times 3)^\color{blue}{5} \\
&= 12^\color{blue}{5} \end{align}||
Impossible to calculate the product (different exponents)
||\begin{align} 2 \times 1.1^3 &= 2^\color{red}{1} \times 1.1^\color{blue}{3} \\
&= 2^\color{red}{1} \times (1.1)^\color{blue}{3} \\
&= 2 \times (1.1)^\color{blue}{3} \end{align}||
As seen in the previous problem, brackets are useful for visually separating two exponential notations that cannot be multiplied.
So, |2 \times 1.1^3 \not= 2.2^3.| The equation remains |2 \times (1.1)^3.|
Example 1 (multiplying powers with the same base)
||\begin{align} &&0.96^{7}&= 0.96^2 \times 0.96^x\\
&&0.96^7 &= 0.96^{2+x} \\
&\Rightarrow &7 &= 2+x \\
&&5 &= x \end{align}||
Example 2 (dividing powers with the same base)
||\begin{align} && \dfrac{1.15^4}{1.15^2} &= 1.15^x \\
&& 1.15^{4-2}&= 1.15^x \\
&& 1.15^2 &= 1.15^x \\
&\Rightarrow &2 &= x \end{align}||
Example 3 (power of a product)
||\begin{align} && 1.2^2 \times 1.4^2 &= x^2 \\
&\Rightarrow &(1.2 \times 1.4)^2 &= x^2 \\
&& 1.68 ^2 &= x^2 \\
&\Rightarrow &1.68 &= x \end{align}||
Example 4 (power of a product)
||\begin{align} &&1.5^3 &= 8 \times 0.75^x \\
&&(2 \times 0.75)^3 &= 8 \times 0.75^x \\
&\Rightarrow & 2^3 \times 0.75^3 &= 8 \times 0.75^x \\
&& 8 \times 0.75^3 &= 8 \times 0.75^x \\
&& 0.75^3 &= 0.75^x \\
&\Rightarrow &3 &= x \end{align}||
Example 5 (power of a power)
||\begin{align}
&&0.7^{2x} &= 0.49^3 \\
&&(\color{blue}{0.7^2})^x &= 0.49^3 \\
&&\color{blue}{0.49}^x &= 0.49^3 \\
&\Rightarrow &x &= 3 \end{align}||
Follow these steps for solving equations.
Isolate |x| as an exponent.
Find equivalent bases.
Obtain powers with the same bases on each side.
Compare the exponents.
The strategy behind this procedure is to use the properties of the exponents to obtain bases with equivalent values. The components can then be compared directly.
Solve the following equation.
||500 (1.4)^{2+x} = 2.45 (28)^2||
Isolate |x| as an exponent
||\begin{align} 500 \color{blue}{(1.4)^{2+x}} &= 2.45 (28)^2 \\
500 \times \color{blue}{1.4^2 \times 1.4^x} &= 2.45 (28)^2 && \text{product of powers} \\
500 \times 1.96 \times 1.4^x &= 2.45 (28)^2 \\
980 (1.4)^x &= 2.45 (28)^2 \end{align}||
Find equivalent bases
||\begin{align}
980 (1.4)^x &= 2.45 (28)^2 \\
980 (\color{blue}{1.4})^x &= 2.45 (20 \times \color{blue}{1.4})^2 && \text{equivalent bases}\\
980 (\color{blue}{1.4})^x &= 2.45 (20)^2 \times (\color{blue}{1.4})^2 &&\text{power of a product} \\ \end{align}||
Obtain powers with the same bases on each side
||\begin{align}
980 (1.4)^x &= 2.45 (20)^2 \times (1.4)^2 \\
980 (1.4)^x &= 2.45 \times 400 \times (1.4)^2 \\
\dfrac{980 (1.4)^x}{\color{red}{980}} &= \dfrac{980 (1.4)^2}{\color{red}{980}} \\
1.4^x &= 1.4^2 \end{align}||
Compare the exponents
||\begin{align}
&&1.4^x &= 1.4^2 \\
&\Rightarrow & x &= 2 \end{align}||
If it is impossible to find the same base for each power, you can use logarithms. Refer to the concept of solving equations involving the laws of logarithms if needed.
These examples use the properties and laws of exponents.
Here is a technique that can be used to simplify an exponential expression.
||\begin{align} (\color{red}{\sqrt [3]{3^{4}}}\times 5^{2})\div \left(\left(\dfrac{1}{5}\right)^{3}\times 3^{^{\frac{1}{3}}}\right) &= (\color{red}{3^{^{\frac{4}{3}}}}\times 5^{2})\div \left(\left(\color{blue}{\dfrac{1}{5}}\right)^{\color{blue}{3}}\times 3^{^{\frac{1}{3}}}\right)\\\\ &= (3^{^{\frac{4}{3}}}\times 5^{2})\div \left(\color{blue}{5^{\text{-}3}}\times 3^{^{\frac{1}{3}}}\right)\\ &= \dfrac{3^{^{\frac{4}{3}}}\times 5^{2}}{\color{blue}{5^{\text{-}3}}\times \color{red}{3^{^{\frac{1}{3}}}}}\\\\ &= \dfrac{3^{^{\frac{4}{3}}}\times 5^{2}}{\color{red}{3^{^{\frac{1}{3}}}}\times \color{blue}{5^{\text{-}3}}}\\\\ &= \dfrac{3^{^{\frac{4}{3}}}}{3^{^{\frac{1}{3}}}}\times \dfrac{5^{2}}{5^{\text{-}3}}\\\\ &=3^{^{\frac{4}{3}\text{-}\frac{1}{3}}}\times 5^{2\text{-}^\text{-}3}\\[4pt] &=3\times 5^{5}\end{align}||
The next example has numbers and variables.
Simplify the following expression and give an answer where the exponents are positive.
||\begin{align}\left(\dfrac{2^{^{\frac{1}{4}}}xy\times \color{red}{4} y}{\color{blue}{4} x^{2}y^{5}}\right)^{\!2}
&=\left(\dfrac{2^{^{\frac{1}{4}}}xy\times \color{red}{2^{2}}y}{\color{blue}{2^{2}}x^{2}y^{5}}\right)^{\!2} && \text{change to base 2}\\\\
&= \dfrac{2^{^{\frac{1}{2}}}x^{2}y^{2}\times 2^{4}y^{2}}{2^{4}x^{4}y^{10}} && \text{distribute the exponent 2}\\\\
&=\dfrac{2^{^{\frac{1}{2}+\,\Large{4}}}x^{2}y^{2+2}}{2^{4}x^{4}y^{10}}&& \text{+ the exponents with the same base}\\\\
&=\dfrac{2^{^{\frac{9}{2}}}x^{2}y^{4}}{2^{4}x^{4}y^{10}}\\\\
&=2^{^{\frac{9}{2}-\,\Large{4}}}x^{2-4}y^{4-10}&& \text{- the exponents with the same base}\\\\
&= 2^{^{\frac{1}{2}}}x^{-2}y^{-6}&&\\\\
&= \dfrac{2^{^{\frac{1}{2}}}}{x^{2}y^{6}}&&\text{transform the - exponents}\\\\
&= \dfrac{\sqrt{2}}{x^{2}y^{6}}&&\text{fractional exponent in radical form}\end{align}||
For the following properties, it is important to remember that
||a^{^{\large{\frac{m}{n}}}} = \sqrt[n]{a^m}|| ||\text{where}\quad a \in \mathbb{R},\quad m \in \mathbb{Z},\quad \text{and}\quad n \in \mathbb{N}^*.||
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Properties |
Examples |
|---|---|
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Multiplying radicals of the same index: |
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Dividing radicals of the same index: |
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Factoring radical: |
||\begin{align} \sqrt[3]{108} &= \sqrt[3]{27 \times 4} \\ &= \sqrt[3]{27} \times\sqrt[3]{4}\\ |
In mathematics, the properties of radicals are mainly used when analyzing the square root function, rationalizing a fraction, and finding the coordinates of points in the unit circle.
Example 1
Simplify |\sqrt{12}.|
||\begin{align}\sqrt{12} &= \sqrt{4 \times 3} \\
&= \sqrt{4} \times \sqrt{3}\\
&=2\sqrt{3}\end{align}||
Example 2
Simplify |\sqrt[3]{16x^4y^2}.|
||\begin{align} \sqrt[3]{\color{blue}{16}\color{red}{x^4}\color{magenta}{y^2}}&= \sqrt[3]{\color{blue}{8 \times 2} \color{red}{ x^3 \times x^1} \times\color{magenta}{y^2}}\\
&=\sqrt[3]{\color{blue}{8}\color{red}{x^3}} \times \sqrt[3]{\color{blue}{2}\color{red}{x^1}\color{magenta}{y^2}}\\
&=\sqrt[3]{\color{blue}{8}} \times\sqrt[3]{\color{red}{x^3}} \times\sqrt[3]{\color{blue}{2}\color{red}{x}\color{magenta}{y^2}}\\
&= \color{blue}{2}\color{red}{x}\sqrt[3]{\color{blue}{2}\color{red}{x}\color{magenta}{y^2}}\\
&= \color{blue}{2}\color{red}{x}({\color{blue}{2}\color{red}{x}\color{magenta}{y^2}})^{^{\frac{1}{3}}}\end{align}||
Example 3
Simplify |\sqrt{\dfrac{36x^3y^4}{5z^6}}.|
||\begin{align}\sqrt{\dfrac{36x^3y^4}{5z^6}}
&=\dfrac{\sqrt{\color{blue}{36}\color{red}{x^3}\color{magenta}{y^4}}}{\sqrt{5z^6}}\\\\
&=\dfrac{\sqrt{\color{blue}{36}\color{red}{x^2\times x}\color{magenta}{y^4}}}{\sqrt{5z^6}}\\\\
&=\dfrac{\sqrt{\color{blue}{36}\color{red}{x^2}\color{magenta}{y^4}}\times\sqrt{\color{red}{x}}}{\sqrt{z^6}\times\sqrt{5}}\\\\
&=\dfrac{\color{blue}{6}\color{red}{x}\color{magenta}{y^2}\times\sqrt{\color{red}{x}}}{z^3\times\sqrt{5}}\\\\
&=\dfrac{\color{blue}{6}\color{red}{x}\color{magenta}{y^2}\times\sqrt{\color{red}{x}}}{z^3\times\sqrt{5}} \times\dfrac{\sqrt{5}}{\sqrt{5}}\\\\
&=\dfrac{\color{blue}{6}\color{red}{x}\color{magenta}{y^2}\times\sqrt{\color{red}{x}}\times \sqrt{5}}{z^3\times \sqrt{5}\times \sqrt{5}} \\\\
&=\dfrac{\color{blue}{6}\color{red}{x}\color{magenta}{y^2}\times\sqrt{\color{red}{x}\times5}}{z^3\times\sqrt{5\times5}} \\\\
&=\dfrac{\color{blue}{6}\color{red}{x}\color{magenta}{y^2}\times\sqrt{5\color{red}{x}}}{z^3\times\sqrt{25}} \\\\
&=\dfrac{\color{blue}{6}\color{red}{x}\color{magenta}{y^2}\sqrt{5\color{red}{x}}}{5z^3}\end{align}||