Question

Evaluate √112-√63+224/√28​

Answer 1

Need To Evaluate :-

• $\sf\: \sqrt{112} - \sqrt{63} + \dfrac{224}{ \sqrt{28} }$

Solution :-

Basic Concept Used :-

• Reduce the terms to simplest form.
• Method of Rationalization :-
• Rationalization is the process of removing radicals from denominator and multiple and divide by the monomial given in denominator.

Given that :-

$\sf :\implies\: \sqrt{112} - \sqrt{63} + \dfrac{224}{ \sqrt{28} } - - - (1)$

Consider:-

• $\green{\sf \: \sqrt{112} = \sqrt{4 \times 4 \times 7} = 4 \sqrt{7}}$
• $\green{\sf \: \sqrt{63} = \sqrt{3 \times 3 \times 7} = 3 \sqrt{7}}$

$\sf :\implies\:\dfrac{224}{\sqrt{28} }=\dfrac{224}{\sqrt{2\times 2\times7}}=\dfrac{224}{2\sqrt{7}} = \dfrac{112}{ \sqrt{7}} \times \dfrac{ \sqrt{7} }{\sqrt{7}} = 16 \sqrt{7}$

• Therefore,

$\sf:\implies\: \sqrt{112} - \sqrt{63} + \dfrac{224}{ \sqrt{28}}$

Substituting all the above values :-

$\sf \: = \: \: 4 \sqrt{7} - 3 \sqrt{7} + 16 \sqrt{7}$

$\sf \: = \: \: \sqrt{7}(4 - 3 + 16)$

$\sf\: = \: \: \sqrt{7}(1+ 16)$

$\sf \red{\: = \: \: 17 \sqrt{7}}$

$\therefore\: \: \pink{ \pink{\boxed{ \sf \:\sqrt{112} - \sqrt{63} + \dfrac{224}{ \sqrt{28} } = 17 \sqrt{7}}}}$

⠀⠀⠀⠀⠀⠀⛦Laws of exponents :-

• $\: \: \: \: \boxed{ \sf\: {a}^{m} \times {a}^{n} = {a}^{m + n}}$

• $\: \: \: \: \boxed{\sf \: {a}^{m} \div {a}^{n} = {a}^{m - n}}$

• $\: \: \: \: \boxed{\sf \: {a}^{0}=1}$

• $\: \: \: \: \boxed{\pmb \: {a}^{ - 1}=\dfrac{1}{a} }$

• $\: \: \: \: \boxed{\sf \: {a}^{ - n}=\dfrac{1}{ {a}^{n}}}$

• $\: \: \: \: \boxed{ \sf \: {( {a}^{n})}^{m}= {a}^{mn} }$