Numbers: Powers and roots
Higher roots
The root we looked at earlier is also known as a square root. This enables us to better distinguish it with the higher roots, which we will define here.
Since #\blue2^\green3=\orange8#, we call #\blue2# the #\green{\text{cube}}# root of #\orange8#. We write this as:
\[\sqrt[\green3]{\orange8}=\blue2\]
Similarly, because #\blue2^\green4=\orange{16}#, we call #\blue2# the #\green{\text{fourth}}# root of #\orange{16}#. We write this as:
\[\sqrt[\green4]{\orange{16}}=\blue2\]
In general we can state:
The #\green{\textit{cube}}# root of a #\orange{\textit{number}}# is a #\blue{\textit{number}}# which, when raised to the power of #\green{3}#, equals #\orange{\textit{the number inside the radical symbol}}#.
The #\green{\textit{fourth}}# root of a #\orange{\textit{number}}# is a #\blue{\textit{non-negative number}}# which, when raised to the power of #\green{4}#, equals #\orange{\textit{the number inside the radical symbol}}#.
We can extend this to even higher roots, noting that if the #\green{\text{index}}# is odd, the #\blue{\text{result}}# can be both positive and negative, and if the #\green{\text{index}}# is even, the #\blue{\text{result}}# can only be non-negative.
Examples
\[\begin{array}{rcl}\sqrt[\green3]{\orange{27}}&=&\blue3 \\ &&\text{because }\blue3^\green{3}=\orange{27} \\ \\\sqrt[\green4]{\orange{625}}&=&\blue5 \\&&\text{because }\blue5^\green4=\orange{625} \text{ and } \blue5 \geq 0 \\ \\\sqrt[\green5]{\orange{32}}&=&\blue2 \\&&\text{because }\blue2^\green5=\orange{32}\\ \\\sqrt[\green3]{\orange{-8}}&=&\blue{-2} \\&&\text{because }(\blue{-2})^\green3=\orange{-8} \\ \\\sqrt[\green6]{\orange{729}}&=&\blue3 \\&&\text{because }\blue3^\green6=\orange{729} \text{ and } \blue3 \geq 0 \end{array}\]
Similar to square roots, higher roots often do not result in integers.
According to the definition of the cube root, it holds that: \[\left(\sqrt[\green3]{\orange2}\right)^3=\orange2\]
We would like to estimate the value of #\sqrt[\green3]{\orange2}#.
Because #1^3=1# and #2^3=8#, we see that #1 \lt \sqrt[\green3]{\orange2} \lt 2#.
Using a calculator, we find the approximation #\sqrt[\green3]{\orange2} \approx 1.25992105....#
Similar to #\sqrt{2}#, #\sqrt[\green3]{\orange2}# has an infinite number of decimal places. When the answer to a question should be exact, #\sqrt[\green3]{\orange2}# is considered a final answer. Like #1#, #\tfrac{1}{2}# #0.6# and #\sqrt{2}#, #\sqrt[\green3]{\orange2}# is a number.
We already saw in the examples in the definition that the cube root of a negative number exists. For higher roots with an even power, this does not hold.
According to the definition of the #\green{\text{fourth}}# root, #\sqrt[\green4]{\orange{-1}}# should be a number that, when raised to the power of #\green4#, equals #\orange{-1}#.
However, when we raise a positive number to power of #\green4#, the result is always a positive number. Therefore, a positive number raised to the power of #\green4# can never be #\orange{-1}#.
A negative number raised to the power of #\green4# also always results in a positive number. Therefore, a negative number raised to the power of #\green4# can never result in #\orange{-1}#.
This means that #\sqrt[\green4]{\orange{-1}}# does not exist. We can only take #\green{\text{fourth}}# roots of non-negative numbers. The same applies to all higher roots with an even #\green{\text{index}}#. For higher roots with an odd #\green{\text{index}}#, this does not hold.
Other examples
\[\begin{array}{rcl}\\ \sqrt[\green3]{\orange{-64}}&=&\blue{-4} \\ \\ \sqrt[\green4]{\orange{-16}} && \text{does not exist} \\ \\ \sqrt[\green5]{\orange{-243}}&=&\blue{-3} \\ \\ \sqrt[\green6]{\orange{-729}} && \text{does not exist}\end{array}\]
When we calculate #\sqrt[5]{1024}#, we are looking for a number which, when raised to the power of #5#, equals #1024#.
In this case:
\[4^5=1024\]
Therefore, #\sqrt[5]{1024}=4#
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