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  • (i) Simplify :- (1power 3 + 2power 3 + 33) power 1 by2

(i) Simplify :- (13 + 23 + 33)1/2
(ii) Simplify :- \left ( \frac{3}{5} \right )^{4}\left ( \frac{8}{5} \right )^{-12}\left ( \frac{32}{5} \right )^{6}
(iii) Simplify :- \left ( \frac{1}{27} \right )^{-\frac{2}{3}}
(iv) Simplify :- \left [ \left \{ \left ( 625 \right )^{-\frac{1}{2}} \right \}^{-\frac{1}{4}} \right ]^{2}
(v) Simplify :- \frac{9^{\frac{1}{3}}\times 27^{\frac{1}{2}}}{3^{\frac{1}{6}}\times 9^{-\frac{2}{3}}}
(vi) Simplify :-64^{-\frac{1}{3}}\left ( 64^{\frac{1}{3}}-64^{\frac{2}{3}} \right )
(vii) Simplify :- \frac{8^{\frac{1}{3}}\times 16^{\frac{1}{3}}}{32^{-\frac{1}{3}}}


 

Answers (1)

(i) Answer.          6
Solution.     (13 + 23 + 33)1/2
We know that
13 = 1.1.1 = 1
23 = 2.2.2 = 8
33 = 3.3.3 = 27
Putting these values we get
(13 + 23 + 33)1/2 = \sqrt{1+8+27}
=\sqrt{36}= 6
Hence the answer is 6

(ii) Answer.  \frac{2025}{64}       

Solution.\left ( \frac{3}{5} \right )^{4}\left ( \frac{8}{5} \right )^{-12}\left ( \frac{32}{5} \right )^{6}
 We know that
8 = 2.2.2 = 23
32 = 2.2.2.2.2 = 25
\left ( \frac{3}{5} \right )^{4}\left ( \frac{8}{5} \right )^{-12}\left ( \frac{32}{5} \right )^{6}= \left ( \frac{3}{5} \right )^{4}\left ( \frac{2^{3}}{5} \right )^{-12}\left ( \frac{2^{5}}{5} \right )^{6}
= \frac{3^{4}\left ( 2^{3} \right )^{-12}\left ( 2^{5} \right )^{6}}{5^{4}5^{-12}5^{6}}                \because \left ( \frac{a}{b} \right )^{m}= \frac{a^{m}}{b^{m}}
= \frac{3^{4}2^{-36}2^{30}}{5^{4}5^{-12}5^{6}}                            \because \left ( \left ( a \right )^{m} \right )^{n}= \left ( a \right )^{mn}
= \frac{3^{4}\times 2^{-36+30}}{5^{4-12+6}}                        \because \left ( a \right )^{m}\left ( a \right )^{n}= \left ( a \right )^{m+n}
= \frac{3^{4}\times 2^{-6}}{5^{-2}}
= \frac{3^{4}\times 5^{2}}{2^{6}}                                   \because \left ( a \right )^{-m}= \left ( \frac{1}{a} \right )^{m}
= \frac{81\times 25}{64}
= \frac{2025}{64}
Hence the answer is \frac{2025}{64}
     
(iii) Answer. 9
Solution. Given \left ( \frac{1}{27} \right )^{-\frac{2}{3}}
We know that
27 = 3.3.3 = 33
\left ( \frac{1}{27} \right )^{-\frac{2}{3}}= \left ( \frac{1}{3^{3}} \right )^{-\frac{2}{3}}
= \left ( 3^{3} \right )^{\frac{2}{3}}            \because \left ( a \right )^{-m}= \left ( \frac{1}{a} \right )^{m}
= \left ( 3 \right )^{3\times \frac{2}{3}}         \because \left ( \left ( a \right )^{m} \right )^{n}= \left ( a \right )^{mn}
= 32 = 9
Hence the answer is 9

(iv) Answer. 5
Solution. Given \left [ \left \{ \left ( 625 \right )^{-\frac{1}{2}} \right \}^{-\frac{1}{4}} \right ]^{2}
We know that
625= \left ( 25 \right )\left ( 25 \right )= 5\cdot 5\cdot 5\cdot 5= 5^{4}
\left [ \left \{ \left ( 625 \right )^{-\frac{1}{2}} \right \}^{-\frac{1}{4}} \right ]^{2}= \left [ \left \{ \left ( \left ( 5 \right )^{4} \right )^{-\frac{1}{2}} \right \} ^{-\frac{1}{4}}\right ]^{2}
= 5^{4\times \frac{-1}{2}\times \frac{-1}{4}\times 2}        \because \left ( \left ( a \right )^{m} \right )^{n}= \left ( a \right )^{mn}

= 51 = 5
Hence the answer is 5

(v) Answer.      \sqrt[3]{\frac{1}{3}}
Solution. We have \frac{9^{\frac{1}{3}}\times 27^{\frac{1}{2}}}{3^{\frac{1}{6}}\times 9^{-\frac{2}{3}}}
Now we know that
9 = 3.3 = 32
27 = 3.3.3 = 33
\frac{9^{\frac{1}{3}}\times 27^{\frac{1}{2}}}{3^{\frac{1}{6}}\times 9^{-\frac{2}{3}}}= \frac{\left ( 3^{2} \right )^{\frac{1}{3}}\times \left ( 3^{3} \right )^{\frac{1}{2}}}{\left ( 3 \right )^{\frac{1}{6}}\times \left ( 3^{2} \right )^{\tfrac{-2}{3}}}
= \frac{\left ( 3\right )^{2\times \frac{1}{3}}\times \left ( 3 \right )^{3\times \frac{-1}{2}}}{\left ( 3 \right )^{\frac{1}{6}}\times \left ( 3 \right )^{\tfrac{-2}{3}}}        \because \left ( \left ( a \right )^{m} \right )^{n}= \left ( a \right )^{mn}
= \frac{\left ( 3 \right )^{\frac{2}{3}-\frac{3}{2}}}{\left ( 3 \right )^{\frac{1}{6}-\frac{2}{3}}}              \because \left ( a \right )^{m}\left ( a \right )^{n}= \left ( a \right )^{m+n}
= \frac{3^{\frac{4-9}{6}}}{3^{\frac{1-4}{6}}}
=\frac{3^{-\frac{5}{6}}}{3^{-\frac{3}{6}}}
= 3^{\frac{-5}{6}-\left ( \frac{-3}{6} \right )}           \because \frac{\left ( a \right )^{m}}{\left ( a \right )^{n}}= \left ( a \right )^{m-n}
= 3^{-\frac{2}{6}}
= \left ( \frac{1}{3} \right )^{\frac{1}{3}}                 \because \left ( a \right )^{-m}= \left ( \frac{1}{a} \right )^{m}
= \sqrt[3]{\frac{1}{3}}
Hence the answer is \sqrt[3]{\frac{1}{3}}

(vi) Answer. – 3
Solution. We have ,64^{-\frac{1}{3}}\left ( 64^{\frac{1}{3}}-64^{\frac{2}{3}} \right )
We know that 64 =4.4.4=43
= \left ( 4^{3} \right )^{\frac{-1}{3}}\left \{ \left ( \left ( 4^{3} \right ) ^{\frac{1}{3}}-\left ( 4^{3} \right ) ^{\frac{2}{3}}\right )\right \}
= \left ( 4 \right )^{3\times \frac{-1}{3}}\left \{ \left ( \left ( 4 \right )^{3\times \frac{1}{3}}-\left ( 4 \right )^{3\times \frac{2}{3}} \right ) \right \}      \because \left ( \left ( a \right )^{m} \right )^{n}= \left ( a \right )^{mn}
= 4^{-1}\left ( 4-4^{2} \right )
= \frac{1}{4}\left ( 4-16 \right )                                                        \because \left ( a \right )^{-m}= \left ( \frac{1}{a} \right )^{m}
= \frac{1}{4}\left ( -12 \right )
= – 3
Hence the answer is – 3
 

(vii) Answer. 16
Solution. Given ,\frac{8^{\frac{1}{3}}\times 16^{\frac{1}{3}}}{32^{-\frac{1}{3}}}
We know that
8 = 2.2.2 = 23
16 = 2.2.2.2 = 24
32 = 2.2.2.2.2 = 25
\frac{8^{\frac{1}{3}}\times 16^{\frac{1}{3}}}{32^{-\frac{1}{3}}}= \frac{\left ( 2^{3} \right )^{\frac{1}{3}}\times\left ( 2^{4} \right )^{\frac{1}{3}}}{\left ( 2^{5} \right )^{-\frac{1}{3}}}
= \frac{2^{3\times\frac{1}{3}}\times2^{4\times\frac{1}{3}}}{2^{5\times\frac{-1}{3}}}              \because \left ( \left ( a \right )^{m} \right )^{n}= \left ( a \right )^{mn}       
= 2^{1+\frac{4}{3}+\frac{5}{3}}                             \because \left ( a \right )^{m}\left ( a \right )^{n}= \left ( a \right )^{m+n}  and
                                                       \because \frac{\left ( a \right )^{m}}{\left ( a \right )^{n}}= \left ( a \right )^{m-n}
= 2^{\frac{3+4+5}{3}}= 2^{\frac{12}{3}}
= 2^{4}= 16
Hence the answer is 16.

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