Question

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

Answer 1

Hint: First, write the numerator and denominator separately in the given fraction. Then, multiply the numerator and denominator with $\sqrt 3 $. Then, combine and simplify the denominator by using the power rule to combine exponents (I). Then, use property (II) to rewrite $\sqrt 3 $ as ${3^{\dfrac{1}{2}}}$ and apply the power rule and multiply exponents (III). Then, simplify the numerator by using the product rule for radicals. We will get the simplified version of $\sqrt {\dfrac{5}{3}} $.

Formula used:
Power rule to combine exponents: ${a^m} \times {a^n} = {a^{m + n}}$
$\sqrt[n]{{{a^x}}} = {a^{\dfrac{x}{n}}}$
${\left( {{a^m}} \right)^n} = {a^{mn}}$

Complete step by step solution:
We have to simplify $\sqrt {\dfrac{5}{3}} $.
So, first rewrite $\sqrt {\dfrac{5}{3}} $ as $\dfrac{{\sqrt 5 }}{{\sqrt 3 }}$.
$ \Rightarrow \dfrac{{\sqrt 5 }}{{\sqrt 3 }}$
Now, multiply numerator and denominator with $\sqrt 3 $.
$ \Rightarrow \dfrac{{\sqrt 5 }}{{\sqrt 3 }} \times \dfrac{{\sqrt 3 }}{{\sqrt 3 }}$
Combine and simplify the denominator.
Multiply $\dfrac{{\sqrt 5 }}{{\sqrt 3 }}$ and $\dfrac{{\sqrt 3 }}{{\sqrt 3 }}$.
$ \Rightarrow \dfrac{{\sqrt 5 \times \sqrt 3 }}{{\sqrt 3 \times \sqrt 3 }}$
Raise $\sqrt 3 $ to the power of $1$.
$ \Rightarrow \dfrac{{\sqrt 5 \times \sqrt 3 }}{{{{\sqrt 3 }^1} \times \sqrt 3 }}$
Raise $\sqrt 3 $ to the power of $1$.
$ \Rightarrow \dfrac{{\sqrt 5 \times \sqrt 3 }}{{{{\sqrt 3 }^1} \times {{\sqrt 3 }^1}}}$
Use the power rule ${a^m} \times {a^n} = {a^{m + n}}$ to combine exponents.
$ \Rightarrow \dfrac{{\sqrt 5 \times \sqrt 3 }}{{{{\sqrt 3 }^{1 + 1}}}}$
Add $1$ and $1$.
$ \Rightarrow \dfrac{{\sqrt 5 \times \sqrt 3 }}{{{{\sqrt 3 }^2}}}$
Rewrite ${\sqrt 3 ^2}$ as $3$.
Use property $\sqrt[n]{{{a^x}}} = {a^{\dfrac{x}{n}}}$ to rewrite $\sqrt 3 $ as ${3^{\dfrac{1}{2}}}$.
$ \Rightarrow \dfrac{{\sqrt 5 \times \sqrt 3 }}{{{{\left( {{3^{\dfrac{1}{2}}}} \right)}^2}}}$
Apply the power rule and multiply exponents, ${\left( {{a^m}} \right)^n} = {a^{mn}}$.
$ \Rightarrow \dfrac{{\sqrt 5 \times \sqrt 3 }}{{{3^{\dfrac{1}{2} \times 2}}}}$
Multiply $\dfrac{1}{2}$ and $2$, we get
$ \Rightarrow \dfrac{{\sqrt 5 \times \sqrt 3 }}{{{3^1}}}$
It can be written as
$ \Rightarrow \dfrac{{\sqrt 5 \times \sqrt 3 }}{3}$
Now, simplify the numerator.
For this combine using the product rule for radicals.
$ \Rightarrow \dfrac{{\sqrt {5 \times 3} }}{3}$
Multiply $5$ by $3$, we get
$ \Rightarrow \dfrac{{\sqrt {15} }}{3}$
The result can be shown in multiple forms.
Exact Form: $\dfrac{{\sqrt {15} }}{3}$
Decimal Form: $1.290994449$

Hence, simplified version of $\sqrt {\dfrac{5}{3}} $ is $\dfrac{{\sqrt {15} }}{3}$.

Note: By simplifying a fraction, we mean to express the fraction as a ratio of prime numbers or we can say that both the numerator and denominator should be prime numbers, that is, they should be divisible by only $1$ and itself. For simplifying a fraction, we write it as a product of prime factors, and then divide both of them with the common factors, in this question both the numerator and denominator are already prime numbers and thus the fraction cannot be simplified further.