Question

(27/8)^1/3×[243/32)^1/5÷(2/3)^2]
Simplify this question sir pleasehelpme​

Answer 1

Explanation:

`= {( (27)/(8) )}^{ (1)/(3) } * ( (243)/(32) )^{ (1)/(5) } / {( (2)/(3) )}^(2)`

`= { ({ ((3)/(2) )}^(3)) }^{ (1)/(3) } * {( {( (3)/(2)) }^(5) )}^{ (1)/(5) } / {( (2)/(3) )}^(2)`

`= {( (3)/(2) )}^{3 * (1)/(3) } * {( (3)/(2) )}^{5 * (1)/(5) } * {( (3)/(2) )}^(2)`

`= (3)/(2) * (3)/(2) * {( (3)/(2) )}^(2)`

`= {( (3)/(2) )}^(1 + 1 + 2)`

`= {( (3)/(2) )}^(4) \: or \: (81)/(16)`

Answer 2

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

`\sf{\longmapsto{\bigg( (27)/(8) \bigg)^{(1)/(3)} * \Bigg[\bigg( (243)/(32) \bigg)^{(1)/(5)} / \bigg((2)/(3) \bigg)^(2)\Bigg]}} \\`

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

`\sf{\longmapsto{\bigg( (3 * 3 * 3)/(2 * 2 * 2) \bigg)^{(1)/(3)} * \Bigg[\bigg( (3 * 3 * 3 * 3 * 3)/(2 * 2 * 2 * 2 * 2) \bigg)^{(1)/(5)} / \bigg((2)/(3) \bigg)^(2)\Bigg]}} \\`

`\sf{\longmapsto{\bigg( \frac{{(3)}^(3)}{{(2)}^(3)} \bigg)^{(1)/(3)} * \Bigg[\bigg( \frac{({3}^(5))}{{(2)}^(5)} \bigg)^{(1)/(5)} / \bigg((2)/(3) \bigg)^(2)\Bigg]}} \\`

Now, we can write as :

(3³/2³) = (3/2)³

(3⁵/2⁵) = (3/2)⁵

`\sf{\longmapsto{\left\{\bigg((3)/(2) \bigg)^(3) \right\}^{(1)/(3)} * \Bigg[\left\{\bigg((3)/(2) \bigg)^(5) \right\}^{(1)/(5)} / \bigg((2)/(3) \bigg)^(2)\Bigg]}} \\`

Now using law of exponent :

`{\sf{({a}^(m))^(n) = {a}^(mn)}}`

`\sf{\longmapsto{\bigg( (3)/(2) \bigg)^{3 * (1)/(3)} * \Bigg[\bigg((3)/(2) \bigg)^{5 * (1)/(5)} / \bigg((2)/(3) \bigg)^(2)\Bigg]}} \\`

`\sf{\longmapsto{\bigg( (3)/(2) \bigg)^{(3)/(3)} * \Bigg[\bigg((3)/(2) \bigg)^{(5)/(5)} / \bigg((2)/(3) \bigg)^(2)\Bigg]}} \\`

`\sf{\longmapsto{\bigg( (3)/(2) \bigg)^(1) *\Bigg[\bigg((3)/(2) \bigg)^(1) / \bigg((2)/(3) \bigg)^(2)\Bigg]}} \\`

`\sf{\longmapsto{\bigg( (3)/(2) \bigg)^(1) * \Bigg[\bigg((3)/(2) \bigg)^(1) * \bigg((3)/(2) \bigg)^(2)\Bigg]}} \\`

`\sf{\longmapsto{\bigg( (3)/(2) \bigg)^(1) * \Bigg[\bigg((3)/(2) \bigg)^(1) * \bigg((3)/(2) * (3)/(2) \bigg)\Bigg]}} \\`

`\sf{\longmapsto{\bigg( (3)/(2) \bigg)^(1) * \Bigg[\bigg((3)/(2) \bigg)^(1) * \bigg((3 * 3)/(2 * 2)\bigg)\Bigg]}} \\`

`\sf{\longmapsto{\bigg( (3)/(2) \bigg)^(1) * \Bigg[\bigg((3)/(2) \bigg)^(1) * \bigg((9)/(4)\bigg)\Bigg]}} \\`

`\sf{\longmapsto{\bigg( (3)/(2) \bigg)* \Bigg[\bigg((3)/(2) \bigg)* \bigg((9)/(4)\bigg)\Bigg]}} \\`

`\sf{\longmapsto{\bigg( (3)/(2) \bigg)* \Bigg[ \: \: (3)/(2) * (9)/(4) \: \: \Bigg]}}\\`

`\sf{\longmapsto{\bigg( (3)/(2) \bigg)* \Bigg[ \: \: (3 * 9)/(2 * 4) \: \: \Bigg]}} \\`

`\sf{\longmapsto{\bigg((3)/(2) \bigg)* \Bigg[ \: \: (27)/(8) \: \: \Bigg]}} \\`

`\sf{\longmapsto{(3)/(2) * (27)/(8)}} \\`

`\sf{\longmapsto{(3 * 27)/(2 * 8)}} \\`

`\sf{\longmapsto{(81)/(16)}\: ≈ \:5.0625\:\red{Ans.}} \\`