Question

Q1.Simplify:

Answer 1

$\textbf{(i)}\\\\\{1^3 +2^3 \} \times \left(\dfrac 13 \right)^2\\\\=(1+8) \left(\dfrac 19 \right)\\\\=9\left(\dfrac 19 \right)\\\\=1$

$\textbf{(ii)}\\\\\{5^{-1}\times 4^{-1}\}^2\\\\=\left(5^{-1} \right)^2 \times \left(4^{-1} \right)^2~~~~~~~~~~~;[(ab)^m = a^mb^m]\\\\=5^{-2}\times 4^{-2}~~~~~~~~~~~~~~~~~~~~;[(a^m)^n = a^{mn}]\\\\=\dfrac 1{5^2} \times \dfrac 1{4^2}~~~~~~~~~~~~~~~~~~~~~~;\left[a^{-m} = \dfrac 1{a^m},~ a\neq 0 \right]\\\\=\dfrac{1}{400}\\\\=0.0025$

$\textbf{(iii)}\\\\\left\{ \left(\dfrac 13 \right)^{-2} \times \left( \dfrac 12 \right)^{-2}\right\}\div \left(\dfrac 14 \right)^{-3}\\\\\\=\left( \dfrac 13 \times \dfrac 12 \right)^{-2} \div\left(4^{-1}\right)^{-3}\\\\\\=\left(\dfrac 16 \right)^{-2} \div 4^3\\\\\\=\left(6^{-1} \right)^{-2}\div 4^3\\\\\\=6^2 \div 4^3\\\\\\=36\div 64\\\\\\=\dfrac{9}{16}\\\\\\=0.5625$

Answer 2

$\purple{\maltese}\large\underline{\underline{\red{\sf\:\: Question :}}}$

Q1.Simplify:

$\\ \sf \: 1) \: \: \: ( \: 1 ^{3} +2 ^{3} \: ) \: × {\bigg( \frac{1}{3} \bigg) }^{2}$

$\sf \: 2) \: \: \: {{( \: \: 5 ^{ - 1} ×4^{ - 1} } } \: ) \: ^{2}$

$\sf \: 3) \: \: \: \bigg(\bigg( \frac{1}{3}\bigg) ^{-1}×\bigg( \frac{1}{2} \bigg)^{-2} \bigg)÷\bigg( \frac{1}{4} \bigg)^{-3}$

$\\\purple{\maltese}\large\underline{\underline{\red{\sf\:\: Solution :}}}$

$\\ \bf \: 1) \: \: \: ( \: 1 ^{3} +2 ^{3} \: ) \: × {\bigg( \frac{1}{3} \bigg) }^{2}$

$\\ \sf \implies\:\: \: \: ( \: 1 ^{3} +2 ^{3} \: ) \: × {\bigg( \frac{1}{3} \bigg) }^{2}$

$\\ \sf \implies\:\: \: \: ( \: 1 + 8\: ) \: × {\bigg( \frac{1}{3} \bigg) }^{2}$

$\\ \sf \implies\:\: \: \: ( \: 1 + 8\: ) \: × {\bigg( \frac{1}{9} \bigg) }$

$\\ \sf \implies\:\: \: \: ( \: 9\: ) \: × {\bigg( \frac{1}{9} \bigg) }$

$\\ \sf \implies\:\: \: \: ( \: \cancel{9}\: ) \: × {\bigg( \frac{1}{\cancel{9}} \bigg) }$

$\\ \sf \implies\:\: \: \: 1$

$\implies\sf \underline{ \boxed{\sf \: \: \: \: 1 \: \: \: }} \: \: \bigstar$

$\\\\\qquad\rule{150pt}{2pt}$

$\bf \: 2) \: \: \: {{( \: \: 5 ^{ - 1} ×4^{ - 1} } } \: ) \: ^{2}$

$\\ \sf \implies\:\: \: \: {{( \: \: 5 ^{ - 1} ×4^{ - 1} } } \: ) \: ^{2}$

$\\ \sf \implies\:\: \: \: {{ {\bigg( \: \: \frac{1}{5} × \frac{1}{4} } } \: \bigg) } ^{2}$

$\\ \sf \implies\:\: \: \: {{ {\bigg( \: \: \frac{1}{20} } } \: \bigg) } ^{2}$

$\\ \sf \implies\:\: \: \: {{ {\bigg( \: \: \frac{1}{400} } } \: \bigg) }$

$\implies\sf \underline{ \boxed{\sf \: \: \: \: \frac{1}{400} \: \: \: }} \: \: \bigstar$

$\\\\\qquad\rule{150pt}{2pt}$

$\\ \bf \: 3) \: \: \: \bigg(\bigg( \frac{1}{3}\bigg) ^{-1}×\bigg( \frac{1}{2} \bigg)^{-2} \bigg)÷\bigg( \frac{1}{4} \bigg)^{-3}$

$\\ \sf \implies\:\: \: \: \bigg(\bigg( \frac{1}{3}\bigg) ^{-1}×\bigg( \frac{1}{2} \bigg)^{-2} \bigg)÷\bigg( \frac{1}{4} \bigg)^{-3}$

$\\ \sf \implies\:\: \: \:(\:\: 3^{2}- 2^{3} \:\:)÷4^{3}$

$\\ \sf \implies\:\: \: \:(\:\: 27- 8\:\:)÷64$

$\\ \sf \implies\:\: \: \:19÷64$

$\\ \sf \implies\:\: \: \:\frac{19}{64}$

$\implies\sf \underline{ \boxed{\sf \: \: \: \: \frac{19}{64} \: \: \: }} \: \: \bigstar$

$\\\\\qquad\rule{150pt}{2pt}$