घातांक एवं करणी भाग-2 (indices and surds)

- June 04, 2019

$sqrt[{4}] {3},, sqrt[{5}] {4},, sqrt[{10}] {12},, 1$ 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- $frac {{(243)}^{frac {n} {5}}cdot {3}^{2n+1}} {{9}^{n}cdot {3}^{n-1}}=frac {{3}^{5times frac {n} {5}}cdot {3}^{2n+1}} {{3}^{2n}cdot {3}^{n-1}}$

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

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

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

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

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

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

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

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

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

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

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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