Question

How do you write \[\sqrt x \] as an exponential form?

Answer 1

Hint: When changing from radical form to fractional exponents, remember the basic forms given as below.
The nth root of a can be written as a fractional exponent with a raised to the reciprocal of that power.
\[\sqrt[n]{a} = {a^{\dfrac{1}{n}}}\]

Complete step by step answer:
It is given that we have to express in\[\sqrt x \] exponential form.
As we know the identity \[\sqrt[n]{a} = {a^{\dfrac{1}{n}}}\] , on comparing we get
\[a = x\] and
\[n = 2\]
Therefore,
\[\sqrt x = {x^{\dfrac{1}{2}}}\]

Note: Fractions are always in the form of \[\dfrac{m}{n}\] where, \[n\] is never zero. If \[n\] becomes zero, the value given by the fraction is undefined.
Exponents are in the form of \[{a^m}\], where \[a\] is the base and is multiplied \[m\] with itself times and \[m\] is called the exponent.
Square roots, which use the radical symbol, are non-binary operations — operations which involve just one number — that ask you, “What number times itself gives you this number under the radical?” To convert the square root to an exponent, you use a fraction in the power to indicate that this stands for a root or a radical.
When changing from radical form to fractional exponents, remember these basic forms:
When the nth root of
\[{a^m}\]
is taken, it’s raised to the 1/n power. When a power is raised to another power, you multiply the powers together, and so the m (otherwise written as m/1) and the 1/n are multiplied together.
\[\sqrt[n]{m} = {a^{\dfrac{m}{n}}}\]
When raising a power to a power, you multiply the exponents, but the bases have to be the same. Because raising a power to a power means that you multiply exponents (as long as the bases are the same)
Here are some examples of changing radical forms to fractional exponents:
\[\sqrt x = {x^{\dfrac{1}{2}}}\]
\[\sqrt[3]{x} = {x^{\dfrac{1}{3}}}\]