Question

Find the square root of $\sqrt {48} - \sqrt {45} $.

Answer 1

Hint: We will do prime factorization of value $48$ and $45$ separately and then we will use the formula of ${\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab$ to get the desired result.


Complete step by step solution:
The given values are $\sqrt {48} - \sqrt {45} $
We will do factorization of $\sqrt {48} $ and $\sqrt {45} $separately

$2$	$48$
$2$	$24$
$2$	$12$
$2$	$6$
$3$	$3$
	$1$
$3$	$45$
$3$	$15$
$5$	$5$
	$1$

\[
  48 = 2 \times 2 \times 2 \times 2 \times 3 \\
  48 = {2^2} \times {2^2} \times 3 \\
 \] And \[
  45 = 3 \times 3 \times 5 \\
  45 = {3^2} \times 5 \\
 \]
Then, \[\sqrt {48} - \sqrt {45} = \sqrt {{2^2} \times {2^2} \times 5} - \sqrt {{3^2} \times 5} \]
\[
  \sqrt {48} - \sqrt {45} = 2 \times 2\sqrt 3 - 3\sqrt 5 \\
  \sqrt {48} - \sqrt {45} = 4\sqrt 3 - 3\sqrt 5 \\
 \]
Hence, the answer of $\sqrt {48} - \sqrt {45} = 4\sqrt 3 - 3\sqrt 5 $
Taking $\sqrt 3 $common on the right side, we have,
\[
   = \sqrt 3 \left( {4 - \sqrt 3 \sqrt 5 } \right) \\
   = \dfrac{{\sqrt 3 }}{2}2\left( {4 - \sqrt 3 \sqrt 5 } \right) \\
 \]
$ = \dfrac{{\sqrt 3 }}{2}\left( {8 - 2\sqrt 3 \sqrt 5 } \right)$
We break $8$ into two parts such that it’s both value will be same as the value in root
$ = \dfrac{{\sqrt 3 }}{2}\left[ {5 + 3 - 2\sqrt 3 \sqrt 5 } \right]$
Here, we use the formula,${\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab$
$ = \dfrac{{\sqrt 3 }}{2}\left\{ {{{\left( {\sqrt 5 } \right)}^2} + {{\left( {\sqrt 3 } \right)}^2} - 2\sqrt 3 \sqrt 5 } \right\}$
$ = \dfrac{{\sqrt 3 }}{2}\left\{ {{{\left( {\sqrt 5 + \sqrt 3 } \right)}^2}} \right\}$
So, square root of$\sqrt {48} + \sqrt {45} = \pm {\left( {\dfrac{{\sqrt 3 }}{2}} \right)^{\dfrac{1}{2}}}\sqrt {{{\left( {\sqrt 5 + \sqrt 3 } \right)}^2}} $
$ = \pm {\left( {\dfrac{{\sqrt 3 }}{2}} \right)^{\dfrac{1}{2}}}\left( {\sqrt 5 + \sqrt 3 } \right)$

Note: Students should divide $8$ into two parts such that both values will be the same as the value in root. If we have taken these values accordingly then we can solve them otherwise we will not be able to find the correct answer.