Question

please help me with this ​

Answer 1

Step-by-step explanation:

(i)

(81/16)-¾

= (3/2)⁴ × -¾

= (3/2)-³

= (2/3)³

= (2×2×2/3×3×3)

= 8/27

The multiplicative inverse of 8/27 is 27/8.

(ii)

{(-3/2)-⁴}½

= {(2/-3)⁴}½

= {(2×2×2×2/(-3)×(-3)×(-3)×(-3))}½

= {16/81)½

= (4/9)² × ½

= 4/9

The multiplicative inverse of 4/9 is 9/4.

(iii)

(5/7)-² × (5/7)⁴ + (5/7)³

= (5/7)-²+⁴ + (5/7)³

= (5/7)² + 5/7)³

= 25/49 + 125/343

= 175/343 + 125/343

= 300/343

The multiplicative inverse of 300/343 is 343/300.

Answer 2

Corrected Question :

• (i) ${\sf{\bigg( \dfrac{18}{16} {\bigg)}^{ \dfrac{ - 3}{4} } }}$

• (ii) ${\sf{\bigg[\; \bigg( \dfrac{ - 3}{2} {\bigg)}^{ - 4 } {\bigg]\; }^{ \dfrac{ 1}{2} }}}$

• (iii) ${\sf{\bigg( \dfrac{5}{7} {\bigg)}^{ - 2} \times \bigg( \dfrac{5}{7} {\bigg)}^{4} + \bigg( \dfrac{5}{7} {\bigg)}^{3} }}$

A N S W E R :

• The multiplicative inverse of ${\sf{\dfrac{8}{27}}}$

• The multiplicative inverse of ${\sf{\dfrac{4}{9}}}$

• The multiplicative inverse of ${\sf{\dfrac{300}{343}}}$

E X P L A N A T I O N :

$\begin{gathered}\begin{gathered}\begin{gathered}\large{\underline {\underline {\sf(\:i \:)}}}\\\end{gathered}\end{gathered}\end{gathered}$

• ${\sf{\bigg( \dfrac{18}{16} {\bigg)}^{ \dfrac{ - 3}{4} } }}$

${\sf{\bigg(\dfrac{3}{2}\bigg)^{4}\:\times\:-\:¾}}$

${\sf{\bigg(\dfrac{3}{2}\bigg)^{- 3}}}$

${\sf{\bigg(\dfrac{2}{3}\bigg)^{ 3}}}$

${\sf{\bigg(\dfrac{2\:\times\:2\:\times\:2}{3\:\times\:3\:\times\:3}\bigg)}}$

:$\implies$${\underline{\boxed{\frak{\purple{\dfrac{8}{27}}}}}}$★

${\sf{The\: multiplicative\; inverse\:of \; \bf{\dfrac{8}{27}} \: \sf{is} \: \bf{\dfrac{27}{8}}.}}$

$~~~~~~~~~~~~~~$ _______________________

$\begin{gathered}\begin{gathered}\begin{gathered}\large{\underline {\underline {\sf(\:ii \:)}}}\\\end{gathered}\end{gathered}\end{gathered}$

• ${\sf{\bigg[\; \bigg( \dfrac{ - 3}{2} {\bigg)}^{ - 4 } {\bigg]\; }^{ \dfrac{ 1}{2} }}}$

${\sf{\bigg[\; \bigg( \dfrac{ 2}{-3} {\bigg)}^{ 4 } {\bigg]\; }^{ \dfrac{ 1}{2} }}}$

${\sf{\dfrac{\bigg[ \bigg(2\:\times\:2\:\times\:2\:\times\:2\bigg)\bigg]}{\bigg(-3\bigg)\:\times\; \bigg(-3\bigg)\; \:\times\; \bigg(-3\bigg)\; \:\times\; \bigg(-3\bigg)\bigg)\; \bigg]½}}}$

${\sf{\bigg(\dfrac{16}{81}\bigg) \: \dfrac{1}{2}}}$

${\sf{\bigg(\dfrac{4}{9}\bigg)^{2}\:\times\:\dfrac{1}{2}}}$

:$\implies$${\underline{\boxed{\frak{\blue{\dfrac{4}{9}}}}}}$★

${\sf{The\: multiplicative\; inverse\:of \; \bf{\dfrac{4}{9}} \: \sf{is} \: \bf{\dfrac{9}{4}}.}}$

$~~~~~~~~~~~~~~$ _______________________

$\begin{gathered}\begin{gathered}\begin{gathered}\large{\underline {\underline {\sf(\:iii \:)}}}\\\end{gathered}\end{gathered}\end{gathered}$

• ${\sf{\bigg( \dfrac{5}{7} {\bigg)}^{ - 2} \times \bigg( \dfrac{5}{7} {\bigg)}^{4} + \bigg( \dfrac{5}{7} {\bigg)}^{3} }}$

${\sf{\bigg(\dfrac{5}{7}\bigg)^{- 2} \; {+}^{4} \; + \; \bigg(\dfrac{5}{7}\bigg)^3}}$

${\sf{\bigg(\dfrac{5}{7}\bigg)^2 \: + \: \bigg(\dfrac{5}{7}\bigg)^3}}$

${\sf{\dfrac{25}{49} \: + \: \dfrac{125}{343}}}$

${\sf{\dfrac{175}{343} \: + \: \dfrac{125}{343}}}$

:$\implies$${\underline{\boxed{\frak{\pink{\dfrac{300}{343}}}}}}$★

${\sf{The\: multiplicative\; inverse\:of \; \bf{\dfrac{300}{343}} \: \sf{is} \: \bf{\dfrac{343}{300}}.}}$

Hence,

• $\underline{\sf{The\: multiplicative\; inverse\:of \; \bf{\dfrac{8}{27}} \: \sf{is} \: \bf{\dfrac{27}{8}}.}}$

• $\underline{\sf{The\: multiplicative\; inverse\:of \; \bf{\dfrac{4}{9}} \: \sf{is} \: \bf{\dfrac{9}{4}}.}}$

• $\underline{\sf{The\: multiplicative\; inverse\:of \; \bf{\dfrac{300}{343}} \: \sf{is} \: \bf{\dfrac{343}{300}}.}}$

$~~~~$$\qquad\quad\therefore{\underline{\textsf{\textbf{Hence, Proved!}}}}$

$~~~~~~~~~~~~~~$ _______________________