Question

$\sf \footnotesize(27/8)^ \frac{1}{3} \times [243/32)^ \frac{1}{5} ÷( \frac{2}{3} )^2$


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Answer 1

$\large\underline{\sf{Solution-}}$

${ \qquad{ \rule{129pt}{2pt}}}$

$\begin{gathered}\sf{\longmapsto{\bigg( \dfrac{27}{8} \bigg)^{\frac{1}{3}} \times \Bigg[\bigg( \dfrac{243}{32} \bigg)^{\frac{1}{5}} \div \bigg(\dfrac{2}{3} \bigg)^{2}\Bigg]}} \\\end{gathered}$

We can write as :

27 = 3 × 3 × 3 = 3³

8 = 2 × 2 × 2 = 2³

243 = 3 × 3 × 3 × 3 × 3 = 3⁵

32 = 2 × 2 × 2 ×2 × 2 = 2⁵

• Now, we can write as :

(3³/2³) = (3/2)³

(3⁵/2⁵) = (3/2)⁵

${ \qquad{ \rule{129pt}{2pt}}}$

$\begin{gathered}\sf{\longmapsto{\left\{\bigg(\frac{3}{2} \bigg)^{3} \right\}^{\frac{1}{3}} \times \Bigg[\left\{\bigg(\frac{3}{2} \bigg)^{5} \right\}^{\frac{1}{5}} \div \bigg(\dfrac{2}{3} \bigg)^{2}\Bigg]}} \\\end{gathered}$

${\sf \footnotesize{\: ⟼{(23)3}31×[{(23)5}51÷(32)2]}}$

Now using law of exponent :

${ \qquad{ \rule{129pt}{2pt}}}$

$\begin{gathered}\sf{\longmapsto{\bigg( \frac{3}{2} \bigg)^{3 \times \frac{1}{3}} \times \Bigg[\bigg(\frac{3}{2} \bigg)^{5 \times \frac{1}{5}} \div \bigg(\dfrac{2}{3} \bigg)^{2}\Bigg]}} \\\end{gathered}$

$\begin{gathered} \sf{\longmapsto{\bigg( \frac{3}{2} \bigg)^{\frac{3}{3}} \times \Bigg[\bigg(\frac{3}{2} \bigg)^{\frac{5}{5}} \div \bigg(\dfrac{2}{3} \bigg)^{2}\Bigg]}} \\\end{gathered}$

$\begin{gathered}\sf{\longmapsto{\bigg( \frac{3}{2} \bigg)^{1} \times\Bigg[\bigg(\frac{3}{2} \bigg)^{1} \div \bigg(\dfrac{2}{3} \bigg)^{2}\Bigg]}} \\\end{gathered}$

$\begin{gathered}\sf{\longmapsto{\bigg( \frac{3}{2} \bigg)^{1} \times \Bigg[\bigg(\frac{3}{2} \bigg)^{1} \times \bigg(\dfrac{3}{2} \bigg)^{2}\Bigg]}} \\\end{gathered}$

• (23)1×[(23)1×(2×23×3)]
• (23)1×[(23)1×(49)]
• (23)×[(23)×(49)]
• (23)×[23×49]
• (23)×[2×43×9]
• 23×827
• 2×83×27
• \sf{1681≈5.0625Ans.}

Therefore,

• 1681≈5.0625Ans.

${ \pink{ \rule{12649pt}{9pt}}}$

Answer 2

System of equations

The followings are the some identities that can he used to find the solution:

$\boxed{\begin{array}{l} \bullet \; \; \left(\dfrac{a}{b}\right)^{m} = \dfrac{a^m}{b^m} \\ \\ \bullet \; \; \: \dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{d}{c} \end{array}}$

where, a, b, c, d are integer and m is the power respectively.

According to the given question, we have been given that,

$\rm\implies\Bigg[{\bigg( \dfrac{27}{8} \bigg)^{\frac{1}{3}} \times \bigg( \dfrac{243}{32} \bigg)^{\frac{1}{5}} \div \bigg(\dfrac{2}{3} \bigg)^{2}\Bigg]}$

Let us frame the given equation and understanding the steps to get our final result.

$\begin{array}{l}\rm\implies\Bigg[{\bigg( \dfrac{27}{8} \bigg)^{\frac{1}{3}} \times \bigg( \dfrac{243}{32} \bigg)^{\frac{1}{5}} \div \bigg(\dfrac{2}{3} \bigg)^{2}\Bigg]} \\ \\ \rm\implies\Bigg[{\dfrac{27^{\frac{1}{3}}}{8^{\frac{1}{3}}} \times \dfrac{243^{\frac{1}{5}}}{32^{\frac{1}{5}}} \div \dfrac{2^{2}}{3^{2}}\Bigg]} \\ \\ \rm\implies\Bigg[{\dfrac{3}{2} \times \dfrac{243^{\frac{1}{5}}}{32^{\frac{1}{5}}} \div \dfrac{2^{2}}{3^{2}}\Bigg]} \\ \\ \rm\implies\Bigg[{\dfrac{3}{2} \times \dfrac{3}{2} \div \dfrac{2^{2}}{3^{2}}\Bigg]} \\ \\ \rm\implies\Bigg[{\dfrac{3}{2} \times \dfrac{3}{2} \div \dfrac{4}{9}\Bigg]} \\ \\ \rm\implies\Bigg[\dfrac{3 \times 3}{2 \times 2} \div \dfrac{4}{9}\Bigg] \\ \\ \rm\implies\Bigg[\dfrac{9}{4} \div \dfrac{4}{9}\Bigg] \\ \\ \rm\implies\Bigg[\dfrac{9}{4} \times \dfrac{9}{4}\Bigg] \\ \\ \implies \dfrac{9 \times 9}{4 \times 4} \\ \\ \implies \dfrac{81}{16} \\ \\ \implies 5 \dfrac{1}{16} \\ \\ \implies \boxed{5.0625}\end{array}$

Hence, the correct answer is 5.0625.