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How do you simplify $\dfrac{5}{\sqrt{3}}$?

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Last updated date: 29th Feb 2024
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IVSAT 2024
Answer
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Hint: In this problem we have to simplify the given value. We can observe that the given fraction value has a root value in the denominator. In order to simplify the given value, we will do rationalization for the given fraction. We know that rationalization involves multiplying the dividing root value which is in the denominator with the same value. So here we will multiply and divide the given fraction with $\sqrt{3}$, and simplify the equation by using the known exponential formula $\sqrt{a}\times \sqrt{a}=a$. Then we will get the simplified solution.

Formula Used:
1. Rationalization of a fraction $\dfrac{a}{\sqrt{b}}$ will be $\dfrac{a}{\sqrt{b}}\times \dfrac{\sqrt{b}}{\sqrt{b}}$.
2. $\sqrt{a}\times \sqrt{a}=a$

Complete step by step solution:
Given that, $\dfrac{5}{\sqrt{3}}$.
We can observe that the given fraction has $\sqrt{3}$ as denominator, so rationalizing the above fraction by multiplying and dividing the given fraction with $\sqrt{3}$ , then we will get
$\dfrac{5}{\sqrt{3}}=\dfrac{5}{\sqrt{3}}\times \dfrac{\sqrt{3}}{\sqrt{3}}$
Simplifying the above fraction by multiplying the numerator with the numerator and denominator with the denominator, then we will get
$\Rightarrow \dfrac{5}{\sqrt{3}}=\dfrac{5\sqrt{3}}{\sqrt{3}\times \sqrt{3}}$
In the above fraction we can observe the value $\sqrt{3}\times \sqrt{3}$ which is similar to $\sqrt{a}\times \sqrt{a}$. We have the exponential formula $\sqrt{a}\times \sqrt{a}=a$, from this formula the value of $\sqrt{3}\times \sqrt{3}$ will be $\sqrt{3}\times \sqrt{3}=3$. Substituting this value in the above equation, then we will get
$\Rightarrow \dfrac{5}{\sqrt{3}}=\dfrac{5\sqrt{3}}{3}$

Hence the simplified form of the fraction $\dfrac{5}{\sqrt{3}}$ is $\dfrac{5\sqrt{3}}{3}$.

Note:
In this problem we have directly calculated the value of $\sqrt{3}\times \sqrt{3}$ by using the formula $\sqrt{a}\times \sqrt{a}=a$. You can also multiply the values in the roots and calculate the square root of the product. Mathematically
$\sqrt{3}\times \sqrt{3}=\sqrt{3\times 3}$
We know that $3\times 3=9$ , then we will have
$\sqrt{3}\times \sqrt{3}=\sqrt{9}$
We have the value of $\sqrt{9}=3$ , then we will get
$\sqrt{3}\times \sqrt{3}=3$.