Question

(ii) What
(ii) What is the value of 313 + V3?
What is the value of 3/(64) ? ?
(v) What is the value of []
3 ?
Mathematics-IX​

Answer 1

To find :

$\sf \: Value \: of \: \sqrt[3]{ {(64)}^{ - 2} }$

Identity used :

$\sf \bullet \: {a}^{ - m} = \dfrac{1}{ {a}^{m} }$

$\sf \bullet \: \sqrt[m]{a} \: = \: {a}^{ \frac{1}{m} }$

$\sf \bullet \: { ({a}^{m} )}^{ n} = {a}^{(mn)}$

$\sf \bullet \: \: {1}^{m} \: = \: 1$

$\sf \bullet \: \dfrac{ {a}^{m} }{ {b}^{m} } = { \bigg( \dfrac{a}{b} \bigg) }^{m}$

Solution :

$\sf \implies \sqrt[3]{ {(64)}^{ - 2} }$

$\sf \implies \sqrt[3]{ { \bigg( \dfrac{1}{64} \bigg)}^{ 2} }$

$\sf \implies \: { \: \bigg( \dfrac{1}{64} \bigg)}^{ \frac{2}{3} }$

$\sf \implies \: { \: \bigg( \dfrac{1}{(2 \times 2 \times 2 \times 2 \times 2 \times 2)} \bigg)}^{ \frac{2}{3} }$

$\sf \implies \: { \: \bigg( \dfrac{1}{ {2}^{6} } \bigg)}^{ \frac{2}{3} }$

$\sf \implies \: { \: \bigg( \dfrac{ {1}^{6} }{ {2}^{6} } \bigg)}^{ \frac{2}{3} }$

$\sf \implies \: { \bigg[ \: { \bigg( \: \dfrac{ {1}}{2 } \bigg) } ^{6} \: \bigg ] }^{ \frac{2}{3} }$

$\sf \implies \: \: { \bigg( \: \dfrac{ {1}}{2 } \bigg) } ^{6 \times \frac{2}{3} }$

$\sf \implies \: \: { \bigg( \: \dfrac{ {1}}{2 } \bigg) } ^{ \frac{12}{3} }$

$\sf \implies \: \: { \bigg( \: \dfrac{ {1}}{2 } \bigg) } ^{4}$

$\sf \implies \: \: \: \dfrac{ {1}^{4} }{2 ^{4 }}$

$\sf \implies \: \: \bold{ \dfrac{ 1}{16}}$

ANSWER : (1/16)

Answer 2

To find :-

${\large{\sqrt[3]{(64)}^{ - 2}}}$

Identify Used :

★ ${\large\rm{ {a}^{ - m} }}$ = ${\sf{\frac{1}{ {a}^{m} }}}$

★ ${\large\rm{\sqrt[m]{a}}}$ = ${\sf{ {a}^{ \frac{1}{m} } }}$

★ ${\large\rm{( {a}^{m} )ⁿ}}$ = ${\sf{{a}^{(mn)} }}$

★ ${\large\rm{{1}^{m} = 1}}$

★ ${\large\rm{\frac{ {a}^{m} }{ {b}^{m} }}}$ = ${\sf{\binom{a}{b}^{m}}}$

Solution :-

$\large\green\implies$ ${\large{\sqrt[3] {(64)}^{ - 2} }}$

$\large\pink\implies$ ${\large{\sqrt[3] \binom{1}{64}²}}$

$\large\green\implies$ ${\large{( \frac{1} {64})^{ \frac{2}{3} } }}$

$\large\pink\implies$ ${\large{( \frac{1}{2 \times 2 \times 2 \times 2 \times 2 \times 2 )})^{ \frac{2}{3} }}}$

$\large\green\implies$ ${\large{\binom{1} {2⁶}^{ \frac{2}{3} } }}$

$\large\pink\implies$ ${\large{\binom{1⁶} {2⁶}^{ \frac{2}{3} } }}$

$\large\green\implies$ ${\large{[( \frac{1}{2} )⁶ {]}^{ \frac{2}{3} } }}$

$\large\pink\implies$ ${\large{( \frac{1}{2} {)}^{6 \times \frac{2}{3} }}}$

$\large\green\implies$ ${\large{( \frac{1}{2} {)}^{ \frac{32}{2} } }}$

$\large\pink\implies$ ${\large{( \frac{1}{2} {)}^{4}}}$

$\large\green\implies$ ${\large{ \frac{1⁴}{2⁴}}}$

$\large\pink\implies$ ${\large\bf{\frac{1}{16}}}$

Hence Proved,

• ${\large\bf{ \binom{1}{16} }}$ $\large{\bf\green{✓}}$