Question

How do you write ${10^{\dfrac{3}{7}}}$ using radical notation?

Answer 1

Hint: A radical notation is an expression containing a square root. If $n$ is a positive integer and $a$ is a real number, then a radical notation can be written as $\sqrt[n]{a}$. The $\sqrt {} $ is known as the radical symbol which is used to denote square roots or ${n^{th}}$roots. $n$ is known as the index and $a$ is known as the radicand which is the number or expression inside the radical symbol. In the question, the given expression is in the exponential form.
On the other hand, the exponential notation expresses the number in the form of ${x^y}$ where $x$ is multiplied by itself $y$ times. Here $x$ is known as the base and $y$ is known as the exponent or power or index.

Complete step by step solution:
An expression in the exponential form can be converted into the radical form by the given relation:
${a^{\dfrac{1}{y}}} = \sqrt[y]{a}$
Now we will raise both the sides of the equation to a power $x$, such that
$ \Rightarrow {({a^{\dfrac{1}{y}}})^x} = {(\sqrt[y]{a})^x}$
From the laws of exponents, we know that ${({b^m})^n} = {b^{mn}}$. On following this law and further simplifying, we get
$ \Rightarrow {a^{\dfrac{x}{y}}} = \sqrt[y]{{{a^x}}}$ where $a$ is the base, $y$ is the index and $x$ is the power to which $a$ is raised.
Now we can use the above expression to solve our question. The given exponential form is ${10^{\dfrac{3}{7}}}$.
Here $a = 10$; $y = 7$ and $x = 3$. Therefore on converting into radical form, we get
${10^{\dfrac{3}{7}}}$$ = \sqrt[7]{{{{10}^3}}}$

Hence, ${10^{\dfrac{3}{7}}}$ can be written in radical notation as $\sqrt[7]{{{{10}^3}}}$.

Note:
The laws of exponents are very useful for solving difficult exponential and radical expressions. Some of them are given below:
1) ${a^m} \times {a^n} = {a^{m + n}}$
2) $\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}$
3) ${({a^m})^n} = {a^{mn}}$