घातांक एवं करणी भाग-2 (indices and surds)

घातांक एवं करणी

$\sqrt[{4}] {3},\, \sqrt[{5}] {4},\, \sqrt[{10}] {12},\, 1$ में सबसे बड़ी संख्या कौन सी है –

हल – $\because \, \sqrt[{4}] {3},\, \sqrt[{5}] {4},\, \sqrt[{10}] {12},\, 1$

$=\, {({3}^{5})}^{1/20},{({4}^{4})}^{1/20},{({12}^{2})}^{1/20},{(1)}^{1/20}$

$=\, {243}^{1/20},{(256)}^{1/20},{(144)}^{1/20},{(1)}^{1/20}$

$\therefore$ सबसे बड़ी संख्या $=\sqrt[{5}] {4}$

2. यदि ${x}^{2}+{y}^{2}+2x+1=0$ है, तो ${x}^{31}+{y}^{35}$ का मान क्या होगा –

हल – हल करने पर, x= -1 व y= 0

$\therefore {x}^{31}+{y}^{35}={(-1)}^{31}{(0)}^{35}=-1$

3. $\frac {{(243)}^{\frac {n} {5}}\cdot {3}^{2n+1}} {{9}^{n}\cdot {3}^{n-1}}$ का मान है –

हल- $\frac {{(243)}^{\frac {n} {5}}\cdot {3}^{2n+1}} {{9}^{n}\cdot {3}^{n-1}}=\frac {{3}^{5\times \frac {n} {5}}\cdot {3}^{2n+1}} {{3}^{2n}\cdot {3}^{n-1}}$

$=\, {3}^{n+2n+1-2n-n+1}$

$=\, {3}^{2}=9$

4. ${(100)}^{1/2}\times {(0.001)}^{1/3}-{(0.0016)}^{1/4}\times {3}^{0}+{(\frac {5} {4})}^{-1}$ सरल करके मान बताइये –

हल – ${(100)}^{1/2}\times {(0.001)}^{1/3}-{(0.0016)}^{1/4}\times {3}^{0}+{(\frac {5} {4})}^{-1}$

$=\, {({10}^{2})}^{1/2}\times {({10}^{-3})}^{1/3}-{({0.2}^{4})}^{1/4}\times 1+\frac {4} {5}$

$=\, 10\times \frac {1} {10}-\frac {1} {5}+\frac {4} {5}$

$=\, 1-\frac {1} {5}+\frac {4} {5}=\frac {8} {5}=1.6$

5. ${(3+2\sqrt {2})}^{-3}+{(3-2\sqrt {2})}^{-2}$ का मान है –

हल – दिए गए व्यंजक $=\, {(\frac {1} {3+2\sqrt {2}})}^{3}+{(\frac {1} {3-2\sqrt {2}})}^{3}$

$=\, {(\frac {3-2\sqrt {2}} {9-8})}^{3}+{(\frac {3+2\sqrt {2}} {9-8})}^{3}$

$=\, [{3}^{3}-(2{\sqrt {2}}^{3}]-3\cdot 3\cdot 2\sqrt {2}(3-2\sqrt {2})$ $+[{3}^{3}+(2{\sqrt {2)}}^{3}]+3\cdot 3\cdot 2\, \sqrt {2}(3+2\sqrt {2})$ $=\, 27-54\sqrt {2}+72+27+54\sqrt {2}+72\, =198$

6. यदि ${3}^{x+8}={27}^{2x+1}$ तब x का मान है –

हल – ${3}^{x+8}={27}^{2x+1}$

${3}^{x+8}={3}^{6x+3}$

$x+8=6x+3$

$x=1$

7. $({16}^{3/2}+{16}^{-3/2})$ का सरलीकृत मान है –

हल – $({16}^{3/2}+{16}^{-3/2})$ $=\, {({4}^{2})}^{3/2}+{({4}^{2})}^{-3/2}$ $=\, {4}^{3}+{4}^{-3}$ $=\, 64+\frac {1} {64}=\frac {4097} {64}$

8. ${(256)}^{0.16}{(256)}^{0.09}$ का मान है –

हल – ${(256)}^{0.16}{(256)}^{0.09}$

$=\, {(256)}^{0.16+0.09}$ $=\, {(256)}^{0.25}$ $=\, {(256)}^{1/4}={({4}^{4})}^{1/4}=4$

9. ${(64)}^{-2/3}\times {(\frac {1} {4})}^{-2}$ का मान है –

हल- ${(64)}^{-2/3}\times {(\frac {1} {4})}^{-2}$ $=\, {({4}^{3})}^{-2/3}\times {({4}^{-1})}^{-2}$ $=\, {4}^{-2}\times {4}^{2}={4}^{0}=1$

10. ${(0.01024)}^{\frac {1} {5}}$ का मान ज्ञात कीजिए –

हल – ${(0.01024)}^{1/5}={(\frac {1024} {100000})}^{1/5}$ $=\, {[{(\frac {4} {10})}^{5}]}^{\frac {1} {5}}=\frac {4} {10}=0.4$

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घातांक एवं करणी भाग-2 (indices and surds)

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