Question

How do you simplify $$\sqrt{{\displaystyle \frac{1}{9}}}$$?

Answer 1

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

Complete step-by-step solution:
We are given the expression $$\sqrt{{\displaystyle \frac{1}{9}}}$$. 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 $$a^{{\displaystyle \frac{1}{2}}}$$, here we have $$a={\displaystyle \frac{1}{9}}$$. By doing this, we get $${\left({\displaystyle \frac{1}{9}}\right)}^{{\displaystyle \frac{1}{2}}}$$. We also know the algebraic property which states that $${\left({\displaystyle \frac{a}{b}}\right)}^m={\displaystyle \frac{a^m}{b^m}}$$, here we have $$a=1$$, $$b=9$$, and $$m={\displaystyle \frac{1}{2}}$$. Using this, we can write the given expression in fraction form as, $${\displaystyle \frac{1^{{\displaystyle \frac{1}{2}}}}{9^{{\displaystyle \frac{1}{2}}}}}$$. 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
$$\Rightarrow {\displaystyle \frac{1^{{\displaystyle \frac{1}{2}}}}{9^{{\displaystyle \frac{1}{2}}}}}={\displaystyle \frac{1}{3}}$$
As $${\displaystyle \frac{1}{3}}$$ cannot be simplified further, this is our answer.
Hence, on simplification, we get that the value of $$\sqrt{{\displaystyle \frac{1}{9}}}$$ equals $${\displaystyle \frac{1}{3}}$$.

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