Question

$\huge \dag \sf{Question}$​

Answer 1

Answer:

Question :

$\sf{:\implies{\bigg\lgroup \dfrac{27}{8} \bigg\rgroup^{\frac{1}{3}} \times \Bigg[\bigg\lgroup \dfrac{243}{32} \bigg\rgroup^{\frac{1}{5}} \div \bigg\lgroup\dfrac{2}{3} \bigg\rgroup^{2}\Bigg]}}$

$\begin{gathered}\end{gathered}$

Solution :

$\begin{gathered}\sf{:\implies{\bigg\lgroup \dfrac{27}{8} \bigg\rgroup^{\frac{1}{3}} \times \Bigg[\bigg\lgroup \dfrac{243}{32} \bigg\rgroup^{\frac{1}{5}} \div \bigg\lgroup\dfrac{2}{3} \bigg\rgroup^{2}\Bigg]}} \\ \end{gathered}$

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

$\begin{gathered}\sf{:\implies{\bigg\lgroup \dfrac{3 \times 3 \times 3}{2 \times 2 \times 2} \bigg\rgroup^{\frac{1}{3}} \times \Bigg[\bigg\lgroup \dfrac{3 \times 3 \times 3 \times 3 \times 3}{2 \times 2 \times 2 \times 2 \times 2} \bigg\rgroup^{\frac{1}{5}} \div \bigg\lgroup\dfrac{2}{3} \bigg\rgroup^{2}\Bigg]}} \\ \end{gathered}$

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

Now, we can write as :

(3³/2³) = (3/2)³

(3⁵/2⁵) = (3/2)⁵

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

Now using law of exponent :

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

$\begin{gathered}\sf{:\implies{\bigg\lgroup \frac{3}{2} \bigg\rgroup^{3 \times \frac{1}{3}} \times \Bigg[\bigg\lgroup\frac{3}{2} \bigg\rgroup^{5 \times \frac{1}{5}} \div \bigg\lgroup\dfrac{2}{3} \bigg\rgroup^{2}\Bigg]}} \\ \end{gathered}$

$\begin{gathered}\sf{:\implies{\bigg\lgroup \frac{3}{2} \bigg\rgroup^{\frac{3}{3}} \times \Bigg[\bigg\lgroup\frac{3}{2} \bigg\rgroup^{\frac{5}{5}} \div \bigg\lgroup\dfrac{2}{3} \bigg\rgroup^{2}\Bigg]}} \\ \end{gathered}$

$\begin{gathered}\sf{:\implies{\bigg\lgroup \frac{3}{2} \bigg\rgroup^{1} \times \Bigg[\bigg\lgroup\frac{3}{2} \bigg\rgroup^{1} \div \bigg\lgroup\dfrac{2}{3} \bigg\rgroup^{2}\Bigg]}} \\ \end{gathered}$

$\begin{gathered}\sf{:\implies{\bigg\lgroup \frac{3}{2} \bigg\rgroup^{1} \times \Bigg[\bigg\lgroup\frac{3}{2} \bigg\rgroup^{1} \times \bigg\lgroup\dfrac{3}{2} \bigg\rgroup^{2}\Bigg]}} \\ \end{gathered}$

$\begin{gathered}\sf{:\implies{\bigg\lgroup \frac{3}{2} \bigg\rgroup^{1} \times \Bigg[\bigg\lgroup\frac{3}{2} \bigg\rgroup^{1} \times \bigg\lgroup\dfrac{3}{2} \times \dfrac{3}{2} \bigg\rgroup\Bigg]}} \\ \end{gathered}$

$\begin{gathered}\sf{:\implies{\bigg\lgroup \dfrac{3}{2} \bigg\rgroup^{1} \times \Bigg[\bigg\lgroup\dfrac{3}{2} \bigg\rgroup^{1} \times \bigg\lgroup\dfrac{3 \times 3}{2 \times 2}\bigg\rgroup\Bigg]}} \\ \end{gathered}$

$\begin{gathered}\sf{:\implies{\bigg\lgroup \dfrac{3}{2} \bigg\rgroup^{1} \times \Bigg[\bigg\lgroup\dfrac{3}{2} \bigg\rgroup^{1} \times \bigg\lgroup\dfrac{9}{4}\bigg\rgroup\Bigg]}} \\ \end{gathered}$

$\begin{gathered}\sf{:\implies{\bigg\lgroup \frac{3}{2} \bigg\rgroup\times \Bigg[\bigg\lgroup\frac{3}{2} \bigg\rgroup \times \bigg\lgroup\dfrac{9}{4}\bigg\rgroup\Bigg]}} \\ \end{gathered}$

$\begin{gathered}\sf{:\implies{\bigg\lgroup \dfrac{3}{2} \bigg\rgroup\times \Bigg[ \: \: \dfrac{3}{2} \times \dfrac{9}{4} \: \: \Bigg]}} \\ \end{gathered}$

$\begin{gathered}\sf{:\implies{\bigg\lgroup \dfrac{3}{2} \bigg\rgroup\times \Bigg[ \: \: \dfrac{3 \times 9}{2 \times 4} \: \: \Bigg]}} \\ \end{gathered}$

$\begin{gathered}\sf{:\implies{\bigg\lgroup \dfrac{3}{2} \bigg\rgroup\times \Bigg[ \: \: \dfrac{27}{8} \: \: \Bigg]}} \\ \end{gathered}$

$\begin{gathered}\sf{ : \implies{\dfrac{3}{2} \times \dfrac{27}{8}}} \\ \end{gathered}$

$\begin{gathered}\sf{ : \implies{\dfrac{3 \times 27}{2 \times 8}}} \\ \end{gathered}$

$\begin{gathered}\sf{ : \implies{\dfrac{81}{16}}} \\ \end{gathered}$

$\bigstar \:\red{\underline{\boxed{\sf{Answer = \dfrac{81}{16}}}}}$

Hence, the answer is 81/16.

$\begin{gathered}\end{gathered}$

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Learn More :

☼ EXPONENT :

↝ The exponent of a number says how many times to use the number in a multiplication.

☼ LAW OF EXPONENT :

The important laws of exponents are given below:

• ↠ ${\rm{{a}^{m} \times {a}^{n} = {a}^{m + n}}}$
• ↠ ${\rm{{a}^{m}/{a}^{n} = {a}^{m - n}}}$
• ↠ ${\rm{({a}^{m})^{n} = {a}^{mn}}}$
• ↠ ${\rm{{a}^{n}/{b}^{n} = ({a/b})^{n} }}$
• ↠ ${\rm{{a}^{0} = 1}}$
• ↠ ${\rm{{a}^{ - m} = {1/a}^{m}}}$
• ↠ ${\rm{{a}^{\frac{1}{n} } = \sqrt[n]{a}}}$

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Answer 2

Explanation:

81/16

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