Question

Can anyone solve this SURDS simplify.​

Answer 1

Given Question :-

$\rm :\longmapsto\:Evaluate : \sqrt{112} - \sqrt{63} + \dfrac{224}{ \sqrt{28} }$

Basic Concept Used :-

1. Reduce the terms to simplest form.

2. Method of Rationalization :-

• Rationalization is the process of removing radicals from denominator and multiple and divide by the monomial given in denominator.

Let's solve the problem now!!

$\green{\large\underline{\sf{Solution-}}}$

Given that

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

Consider,

$\blue{\bf :\longmapsto\: \sqrt{112} = \sqrt{4 \times 4 \times 7} = 4 \sqrt{7}}$

$\green{\bf :\longmapsto\: \sqrt{63} = \sqrt{3 \times 3 \times 7} = 3 \sqrt{7}}$

$\red{\bf :\longmapsto\:\dfrac{224}{\sqrt{28} }=\dfrac{224}{\sqrt{2\times 2\times7}}=\dfrac{224}{2\sqrt{7}} = \dfrac{112}{ \sqrt{7}} \times \dfrac{ \sqrt{7} }{\sqrt{7}} = 16 \sqrt{7} }$

Hence,

$\rm :\longmapsto\: \sqrt{112} - \sqrt{63} + \dfrac{224}{ \sqrt{28}}$

can be rewritten as after substituting all the above values

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

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

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

$\rm \: = \: \: 17 \sqrt{7}$

Therefore,

$\: \: \: \: \: \: \: \: \pink{ \underbrace{\purple{\boxed{ \bf \:\sqrt{112} - \sqrt{63} + \dfrac{224}{ \sqrt{28} } = 17 \sqrt{7}}}}}$

Additional Information :-

Laws of exponents :-

$(1) \: \: \: \: \purple{\boxed{ \bf \: {a}^{m} \times {a}^{n} = {a}^{m + n}}}$

$(2) \: \: \: \: \purple{\boxed{ \bf \: {a}^{m} \div {a}^{n} = {a}^{m - n}}}$

$(3) \: \: \: \: \purple{\boxed{ \bf \: {a}^{0}=1}}$

$(4) \: \: \: \: \purple{\boxed{ \bf \: {a}^{ - 1}=\dfrac{1}{a} }}$

$(5) \: \: \: \: \purple{\boxed{ \bf \: {a}^{ - n}=\dfrac{1}{ {a}^{n}}}}$

$(6) \: \: \: \: \purple{\boxed{ \bf \: {( {a}^{n})}^{m}= {a}^{mn} }}$