Question

How do you simplify \[\sqrt{\dfrac{1}{9}}\]?

Answer 1

Hint: To solve the given question, we should know some of the algebraic properties. The first property, we should know is \[{{\left( \dfrac{a}{b} \right)}^{m}}=\dfrac{{{a}^{m}}}{{{b}^{m}}}\]. We should also know that \[\sqrt{a}\] can also be written as \[{{a}^{\dfrac{1}{2}}}\]. We will use these properties to find the value of the given expression.

Complete step-by-step solution:
We are given the expression \[\sqrt{\dfrac{1}{9}}\]. We need to simplify and find its value. The given expression is of the form \[\sqrt{a}\], we know it can also be written as \[{{a}^{\dfrac{1}{2}}}\], here we have \[a=\dfrac{1}{9}\]. By doing this, we get \[{{\left( \dfrac{1}{9} \right)}^{\dfrac{1}{2}}}\]. We also know the algebraic property which states that \[{{\left( \dfrac{a}{b} \right)}^{m}}=\dfrac{{{a}^{m}}}{{{b}^{m}}}\], here we have \[a=1\], \[b=9\], and \[m=\dfrac{1}{2}\]. Using this, we can write the given expression in fraction form as, \[\dfrac{{{1}^{\dfrac{1}{2}}}}{{{9}^{\dfrac{1}{2}}}}\]. We know that the square root of 1 is 1 itself, and the square root of 9 is 3. Substituting these values in the above expression, we get
\[\Rightarrow \dfrac{{{1}^{\dfrac{1}{2}}}}{{{9}^{\dfrac{1}{2}}}}=\dfrac{1}{3}\]
As \[\dfrac{1}{3}\] cannot be simplified further, this is our answer.
Hence, on simplification, we get that the value of \[\sqrt{\dfrac{1}{9}}\] equals \[\dfrac{1}{3}\].

Note: To solve these types of questions we should know different algebraic properties. For example, those we used in the above solution \[{{\left( \dfrac{a}{b} \right)}^{m}}=\dfrac{{{a}^{m}}}{{{b}^{m}}}\], and \[\sqrt{a}\] can also be expressed as \[{{a}^{\dfrac{1}{2}}}\]. We can also have similar property for radical powers like cube roots \[\sqrt[3]{a}={{a}^{\dfrac{1}{3}}}\]. for general cases, we can say that \[\sqrt[n]{a}\] is also expressed as \[{{a}^{\dfrac{1}{n}}}\].
We can also use this property to express \[\sqrt[n]{\dfrac{a}{b}}\] as \[\dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}\].