Question

How do you simplify (3rd root of 5) divided by (sqrt of 5)?

Answer 1

Hint: This type of problem is based on the concept of properties of powers. First, we have to consider the whole statement and convert them into an expression. The 3rd root of 5 is \[\sqrt[3]{5}\] and the square root of 5 is \[\sqrt{5}\]. Now, we have to divide \[\sqrt[3]{5}\] by \[\sqrt{5}\]. Using the property \[\sqrt[n]{a}={{a}^{\dfrac{1}{n}}}\], we get the expression as \[\dfrac{{{5}^{\dfrac{1}{3}}}}{{{5}^{\dfrac{1}{2}}}}\]. Now, we have to use the division rule of powers, that is \[\dfrac{{{a}^{n}}}{{{a}^{m}}}={{a}^{n-m}}\], to solve the expression further.

Complete step by step solution:
According to the question, we are asked to simplify (3rd root of 5) divided by (sqrt of 5).
We have been given the statement is (3rd root of 5) divided by (sqrt of 5).
First let us consider the 3rd root of 5.
We know that the nth root of a term ‘x’ is written as \[\sqrt[n]{x}\].
Therefore, we express the statement as \[\sqrt[3]{5}\].
Now, let us consider the square root of 5.
We express the statement as \[\sqrt[2]{5}\] which can also be written as \[\sqrt{5}\].
But, the statement says that (3rd root of 5) divided by (sqrt of 5).
Therefore, on expressing as a function, we get
\[\dfrac{\sqrt[3]{5}}{\sqrt{5}}\] -----(1)
We have to now simplify the expression (1).
We know that \[\sqrt[n]{a}={{a}^{\dfrac{1}{n}}}\].
We can write \[\sqrt[3]{5}\] as \[{{5}^{\dfrac{1}{3}}}\] and \[\sqrt{5}\] as \[{{5}^{\dfrac{1}{2}}}\].
On using this property in the expression (1), we get
\[\dfrac{\sqrt[3]{5}}{\sqrt{5}}=\dfrac{{{5}^{\dfrac{1}{3}}}}{{{5}^{\dfrac{1}{2}}}}\]
Let us use the property \[\dfrac{{{a}^{n}}}{{{a}^{m}}}={{a}^{n-m}}\], to simplify the expression further.
Here, we find that a=5, \[n=\dfrac{1}{3}\] and \[m=\dfrac{1}{2}\].
Therefore, we get
\[\dfrac{\sqrt[3]{5}}{\sqrt{5}}={{5}^{\dfrac{1}{3}-\dfrac{1}{2}}}\]
Let us take LCM in the power of 5.
\[\Rightarrow \dfrac{\sqrt[3]{5}}{\sqrt{5}}={{5}^{\dfrac{2-3}{3\times 2}}}\]
On further simplification, we get
\[\dfrac{\sqrt[3]{5}}{\sqrt{5}}={{5}^{\dfrac{2-3}{6}}}\]
\[\Rightarrow \dfrac{\sqrt[3]{5}}{\sqrt{5}}={{5}^{\dfrac{-1}{6}}}\]
We know that \[{{x}^{-n}}=\dfrac{1}{x}\]. Using this property, we get
\[\dfrac{\sqrt[3]{5}}{\sqrt{5}}=\dfrac{1}{{{5}^{\dfrac{1}{6}}}}\]
Since 1 to the power any term is always equal to 1, we can write 1 as \[{{1}^{\dfrac{1}{6}}}\].
\[\Rightarrow \dfrac{\sqrt[3]{5}}{\sqrt{5}}=\dfrac{{{1}^{\dfrac{1}{6}}}}{{{5}^{\dfrac{1}{6}}}}\]
Using the property of division, that is \[\dfrac{{{a}^{n}}}{{{b}^{n}}}={{\left( \dfrac{a}{b} \right)}^{n}}\], we get
\[\dfrac{\sqrt[3]{5}}{\sqrt{5}}={{\left( \dfrac{1}{5} \right)}^{\dfrac{1}{6}}}\]
As mentioned above, we know that \[\sqrt[n]{a}={{a}^{\dfrac{1}{n}}}\]. Using the same property, we can further simplify the expression.
We get
\[\dfrac{\sqrt[3]{5}}{\sqrt{5}}=\sqrt[6]{\dfrac{1}{5}}\]
Therefore, the simplified expression of (3rd root of 5) divided by (sqrt of 5) is \[\sqrt[6]{\dfrac{1}{5}}\].

Note:
We can also write the simplified expression as \[{{\left( \dfrac{1}{5} \right)}^{\dfrac{1}{6}}}\]. We should know the properties of powers to solve these types of questions. Avoid calculation mistakes based on sign convention. Do not get confused with the term sqrt which means square root and not square.