Question

Find the value...

please fast​

Answer 1

Given :

$\sqrt[3]{4 + \sqrt[3]{ \: 61 + \sqrt[3]{ \: 27 \: } } }$

To find :

It's value

Identity used :

$\sf \bullet \: a^m \times b^m = ( a\times b)^m$

$\sf \bullet \: \sqrt[m]{a^m} \:= a$

Solution :

Here we will solve the inner one cube root and then middle and then the 1st one.

$\sf \implies \: \sqrt[3]{4 + \sqrt[3]{ \: 61 + \sqrt[3]{ \: (3 \times 3 \times 3) \: } } }$

$\sf \implies \: \sqrt[3]{4 + \sqrt[3]{ \: 61 + \sqrt[3]{ \: {3}^{3} \: } } }$

$\sf \implies \: \sqrt[3]{4 + \sqrt[3]{ \: 61 + 3 } }$

$\sf \implies \: \sqrt[3]{4 + \sqrt[3]{ \: 64 } }$

$\sf \implies \: \sqrt[3]{4 + \sqrt[3]{ \:(2 \times 2 \times 2 \times 2 \times 2 \times 2) } }$

$\sf \implies \: \sqrt[3]{4 + \sqrt[3]{ \:(2 \times 2 \times 2) \times (2 \times 2 \times 2) } }$

$\sf \implies \: \sqrt[3]{4 + \sqrt[3]{ \: {2}^{3} \times {2}^{3} } }$

$\sf \implies \: \sqrt[3]{4 + \sqrt[3]{ \: (2 \times 2)^{3} } }$

$\sf \implies \: \sqrt[3]{4 + 4 }$

$\sf \implies \: \sqrt[3]{8}$

$\sf \implies \: \sqrt[3]{(2 \times 2 \times 2)}$

$\sf \implies \: \sqrt[3]{ {2}^{3} }$

$\sf \implies \: \bold{2}$

ANSWER : 2

Answer 2

Given :-

• ${\sf{\sqrt[3]{4 \: + \: \sqrt[3]{61 \: + \: \sqrt[3]{27} } }}}$

To find :-

• Find it's value ?

★ $\large\underline{\frak{As \: we \: know\: that,}}$

$\large\dag$ Formula Used :

• $\boxed{\bf{ {a}^{m} \: \times \: {b}^{m} = ( {a}^{m} \: \times \: {b}^{m)} }}$$\large\star$

• $\boxed{\bf{ \sqrt[m]{ {a}^{m} } = a}}$$\large\star$

Solution :-

Now,

We have,

• Solve the inner one cube root and then middle and then the 1st one.

Solving :

:$\implies$${\sf{\sqrt[3]{4 \: + \: \sqrt[3]{16 \: + \: \sqrt[3]{(3 \: \times 3\: \times 3\: ) } } }}}$

$~~~~~$:$\implies$${\sf{\sqrt[3]{4 \: + \: \sqrt[3]{16 \: + \: \sqrt[3]{ {3}^{3} } } } }}$

$~~~~~~~~~~$:$\implies$${\sf{\sqrt[3]{4 \: + \: \sqrt[3]{61 \: + \: 3} } }}$

$~~~~~~~~~~~~~~~$:$\implies$${\sf{\sqrt[3]{4 \: + \: \sqrt[3]{64} }}}$

:$\implies$${\sf{\sqrt[3]{4 \: + \: \sqrt[3]{(2 \times 2 \times 2 \times 2 \times 2 \times 2)} }}}$

:$\implies$${\sf{\sqrt[3]{4 \: + \: \sqrt[3]{(2 \times 2 \times 2) \times (2 \times 2 \times 2)} }}}$

$~~~~~$:$\implies$${\sf{\sqrt[3]{4 \: + \: \sqrt[3]{ {2}^{3} \: \times \: {2}^{3} } }}}$

$~~~~~~~~~~$:$\implies$${\sf{\sqrt[3]{4 \: + \: \sqrt[3]{(2 \: \times \: 2 {)}^{3} } } }}$

$~~~~~~~~~~~~~~~$:$\implies$${\sf{ \sqrt[3]{4 \: + \: 4}}}$

$~~~~~~~~~~~~~~~~~~~~$:$\implies$${\sf{\sqrt[3]{8}}}$

$~~~~~~~~~~~~~~~~~~~~~~~~~$:$\implies$${\sf{\sqrt[3]{(2 \: \times \: 2 \: \times \: 2)} }}$

$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$:$\implies$${\sf{\sqrt[3]{ {2}^{3} }}}$

$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$:$\implies$${\underline{\boxed{\pink{\frak{2}}}}}$★

$\large\dag$ Hence Verified,

• The value is = $\large\underline{\rm{2}}$