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  • (i)Simplify the following : sqroot 45-3sqroot- 20+4sqroot 5

(i) Simplify the following : \sqrt{45}-3\sqrt{20}+4\sqrt{5}
(ii) Simplify the following : \frac{\sqrt{24}}{8}+\frac{\sqrt{54}}{9}
(iii) Simplify the following : 4\sqrt{12}\times 7\sqrt{6}
(iv) Simplify the following : 4\sqrt{28}\div 3\sqrt{7}\div 3\sqrt{7}
(v) Simplify the following : 3\sqrt{3}+2\sqrt{27}+\frac{7}{\sqrt{3}}
(vi) Simplify the following : \left ( \sqrt{3}-\sqrt{2} \right )^{2}
(vii) Simplify the following : \sqrt[4]{81}- 8\sqrt[3]{216}+15\sqrt[5]{32}+\sqrt{255}
(viii) Simplify the following : \frac{3}{\sqrt{8}}+\frac{1}{\sqrt{2}}
(ix) Simplify the following : \frac{2\sqrt{3}}{3}-\frac{\sqrt{3}}{6}






 

Answers (1)

(i) Answer. \sqrt{5}
Solution.    \sqrt{45}-3\sqrt{20}+4\sqrt{5} 
We know that,
45 = 3\times 3\times 5
20 = 2\times 2\times 5
So we get
\sqrt{3\times3\times5 }-3\sqrt{2\times2\times5}+4\sqrt{5}
= 3\sqrt{5}-3\left ( 2\sqrt{5} \right )+4\sqrt{5}
= 3\sqrt{5}-6\sqrt{5}+4\sqrt{5}
= 7\sqrt{5}-6\sqrt{5}
= \sqrt{5}  
Hence the answer is \sqrt{5}

(ii)  Answer. \frac{7\sqrt{6}}{12}
Solution. We have, \frac{\sqrt{24}}{8}+\frac{\sqrt{54}}{9}
We know that,

24= 6\times 4= 3\times 2\times 2\times 2
54= 9\times 6= 3\times 3\times 3\times 2
So we get
\frac{\sqrt{24}}{8}+\frac{\sqrt{54}}{9}= \frac{2\sqrt{6}}{8}+\frac{3\sqrt{6}}{9}
= \frac{\sqrt{6}}{4}+\frac{\sqrt{6}}{3}
Taking LCM (3,4) = 12

= \frac{3\sqrt{6}+4\sqrt{6}}{12}
= \frac{7\sqrt{6}}{12}

(iii) Answer.  \sqrt[28]{2^{18} \times 3^{11}}
Solution.   We have
\sqrt[4]{12}\times \sqrt[7]{6}
We know that
12 = 2\times 2\times 3
6 = 2\times 3
So we get,
=\sqrt[4]{2\times 2\times 3}\times \sqrt[7]{2\times 3}  
=2^{1 / 4} \cdot 2^{1 / 4} \cdot 3^{1 / 4} \cdot 2^{1 / 7} \cdot 3^{1 / 7}
=2^{\frac{1}{4}+\frac{1}{4}+\frac{1}{7}} \times 3^{\frac{1}{4}+\frac{1}{7}}

=2^{9 / 14} \times 3^{11 / 28}

=\sqrt[28]{2^{18} \times 3^{11}}
Hence the number is \sqrt[28]{2^{18} \times 3^{11}}.

(iv)Answer.    \frac{8}{\left ( 3\times \sqrt[3]{7} \right )}     
Solution.   We have, 4\sqrt{28}\div 3\sqrt{7}\div \sqrt[3]{7}
We know that
28 = 4\times 7
So we can write,
4\sqrt{28}\div 3\sqrt{7}\div \sqrt[3]{7}= \left [ \frac{4\sqrt{28}}{3\sqrt{7}} \right ]\div 7^{\frac{1}{3}}
= \left [ \frac{4\sqrt{4\times 7}}{3\sqrt{7}} \right ]\div 7^{\frac{1}{3}}

= \left [ \frac{4\times 2\sqrt{ 7}}{3\sqrt{7}} \right ]\div 7^{\frac{1}{3}}

= \frac{8}{3}\div 7^{\frac{1}{3}}
= \frac{8}{\left ( 3\times 7^{\frac{1}{3}} \right )}
= \frac{8}{\left ( 3\times \sqrt[3]{7} \right )}
  Hence the answer is \frac{8}{\left ( 3\times \sqrt[3]{7} \right )}

(v) Answer.  3\sqrt{3}+2\sqrt{27}+\frac{7}{\sqrt{3}}
We know that
27 = 3\times 3\times 3
So, 3\sqrt{3}+2\sqrt{27}+\frac{7}{\sqrt{3}} = 3\sqrt{3}+2\sqrt{3\times 3\times 3}+\frac{7}{\sqrt{3}}

= 3\sqrt{3}+2\left ( 3\sqrt{3} \right )+\frac{7}{\sqrt{3}}\times \frac{\sqrt{3}}{\sqrt{3}}                
(Rationalising the denominator)
= 3\sqrt{3}+6\left ( \sqrt{3} \right )+\frac{7\sqrt{3}}{3}

= \left ( 3+6+\frac{7}{3} \right )\sqrt{3}         (Taking \sqrt{3} common)
Now LCM (1,1,3) = 3
= \left ( \frac{9+18+7}{3} \right )\sqrt{3}
= \frac{34}{3}\sqrt{3}
= 19\cdot 63
Hence the answer is 19.63

(vi) Answer. 5-2\sqrt{6}
Solution. Given, \left ( \sqrt{3}-\sqrt{2} \right )^{2}
We know that (a + b)2 = a2 – 2ab + b2
Comparing the given equation with the identity, we get:

\left ( \sqrt{3}-\sqrt{2} \right )^{2}= \left ( \sqrt{3} \right )^{2}-2\left ( \sqrt{3} \right )\left ( \sqrt{2} \right )+\left ( \sqrt{2} \right )^2
= 3 + 2 – 2\sqrt{3\times 2}
= 5-2\sqrt{6}
Hence the answer is 5-2\sqrt{6}

(vii) Answer. 0
Solution. We have, \sqrt[4]{81}-8 \sqrt[3]{216}+15\sqrt[5]{32}+\sqrt{225}
We know that
81 = 3\times 3\times3\times3
216 = 6\times 6\times6
32 = 2\times 2\times2\times2\times2
225 = 15\times 15
So,\sqrt[4]{81}-8 \sqrt[3]{216}+15\sqrt[5]{32}+\sqrt{225}
= \sqrt[4]{3\times3\times3\times3 }-8\sqrt[3]{6\times6\times6}+15\sqrt[5]{2\times2\times2\times2\times2}+\sqrt{15\times15}
= 3 – 8 × 6 + 15 × 2 + 15
= 3 – 48 + 30 + 15
= – 45 + 45
= 0
Hence the answer is 0

(viii) Answer.   \frac{5}{2\sqrt{2}}

Solution.   We have, \frac{3}{\sqrt{8}}+\frac{1}{\sqrt{2}}
  We know that, 8 =2\times 2\times 2
So,

\frac{3}{\sqrt{8}}+\frac{1}{\sqrt{2}} = \frac{3}{\sqrt{2\times 2\times 2}}+\frac{1}{\sqrt{2}}

= \frac{3}{2\sqrt{2}}+\frac{1}{\sqrt{2}}

= \frac{3}{2\sqrt{2}}+\frac{2}{2\sqrt{2}}
= \frac{5}{2\sqrt{2}}
Hence the answer is \frac{5}{2\sqrt{2}}

(ix) Answer.     \frac{\sqrt{3}}{2}
Solution.     We have, \frac{2\sqrt{3}}{3}-\frac{\sqrt{3}}{6}
LCM (3,6) = 6

\frac{2\sqrt{3}}{3}-\frac{\sqrt{3}}{6}= \frac{4\sqrt{3}}{6}-\frac{\sqrt{3}}{6}

= \frac{4\sqrt{3}-\sqrt{3}}{6}

= \frac{3\sqrt{3}}{6}
= \frac{\sqrt{3}}{2}
Hence the answer is \frac{\sqrt{3}}{2}.

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