Question

How do you write \[{\left( {125} \right)^{\dfrac{1}{3}}}\] in radical form?

Answer 1

Hint: In this question we have to convert the exponential from to the radical form, Rewrite the expression with the fractional exponent as a radical. The denominator of the fraction determines the root, in this case the cube root. The exponent refers only to the part of the expression immediately to the left of the exponent, in this case 125.

Complete step-by-step solution:
An algebraic expression that contains radicals is called a radical expression. We use the product and quotient rules to simplify them, the radical symbol \[\sqrt[n]{{}}\], denotes the \[{n^{th}}\] root of that digit. Radical symbol is employed to signify one of the two inverse operations for exponentiation. Radicals are generally indicated as fractional powers. It is generally written as \[{\left( x \right)^{\dfrac{1}{n}}}\], where n is called the index and x is called radicand.
To easily simplify a \[{n^{th}}\] root, we can divide the powers by the index, the quotient is the exponent of the factor outside of the radical, and the remainder is the exponent of the factor left inside the radical i.e., \[\sqrt[n]{{{a^x}}} = {a^{\dfrac{x}{n}}}\].
Given expression is \[{\left( {125} \right)^{\dfrac{1}{3}}}\],
Now we have to find the radical form of the exponential form,
If \[n\] is a positive integer that is greater than \[x\] and \[a\] is a real number or a factor, then \[{a^{\dfrac{x}{n}}} = \sqrt[n]{{{a^x}}}\],
Given form is in the cube roots, and it is in form \[{\left( a \right)^{\dfrac{x}{n}}}\] ,
Here \[a = 125,x = 1,n = 3\],
Now substituting the values, we get,
\[{\left( {125} \right)^{\dfrac{1}{3}}} = \sqrt[3]{{{{125}^1}}}\],
The radical form of the given expression is\[\sqrt[3]{{125}}\] .
Here 125 can be written as \[125 = 5 \times 5 \times 5\],
Which can be again written as \[5 \times 5 \times 5 = {5^3}\],
So here the given expression can be rewritten as \[{\left( {{5^3}} \right)^{\dfrac{1}{3}}}\],
Here we have to divide the power by the index, we get,
\[{\left( {{5^3}} \right)^{\dfrac{1}{3}}} = {5^{\dfrac{3}{3}}}\],
Now simplifying we get,
\[ \Rightarrow {5^{\dfrac{3}{3}}} = 5\],
5 is the decimal form of the given expression.
The radical form of the given expression is \[\sqrt[3]{{125}}\] .

\[\therefore \]The radical form of the expression \[{\left( {125} \right)^{\dfrac{1}{3}}}\] is \[\sqrt[3]{{125}}\] which is also equal to $5$.

Note: A radical symbol \[\sqrt {} \] is used to represent a radical expression but, many students misguidedly read this a square root symbol and multiple times it employed to conclude the square root of a number. However apart from square roots, it can also be used to denote cube root, a fourth root, or higher roots with numbers written in its place accordingly, but in case of square root that no number is written over the radical symbol.