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# How do you simplify $\sqrt{\dfrac{100}{9}}$?

Last updated date: 13th Jun 2024
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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.
We are given the expression $\sqrt{\dfrac{100}{9}}$. 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 $a=\dfrac{100}{9}$. By doing this, we get ${{\left( \dfrac{100}{9} \right)}^{\dfrac{1}{2}}}$. We also know the algebraic property which states that ${{\left( \dfrac{a}{b} \right)}^{m}}=\dfrac{{{a}^{m}}}{{{b}^{m}}}$, here we have $a=100$, $b=9$, and $m=\dfrac{1}{2}$. Using this, we can write the given expression in fraction form as, $\dfrac{{{100}^{\dfrac{1}{2}}}}{{{9}^{\dfrac{1}{2}}}}$. We know that the square root of 100 is 10, and the square root of 9 is 3. Substituting these values in the above expression, we get
$\Rightarrow \dfrac{{{100}^{\dfrac{1}{2}}}}{{{9}^{\dfrac{1}{2}}}}=\dfrac{10}{3}$
As $\dfrac{10}{3}$ cannot be simplified further, this is our answer.
Hence, on simplification, we get that the value of $\sqrt{\dfrac{100}{9}}$ equals $\dfrac{10}{3}$.
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 $\dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}$.