Question

value of expression
$log_{2}( \sqrt[5]{2 \times \sqrt[3]{2 \sqrt{2} } } )$
is:​

Answer 1

Given :

log₂ [  ⁵√{ 2 × ∛( 2√2 ) ]

Note : we will use ⁿ√a = a^( 1/n ) more frequently while solving this question

By simplifying the given expression we get ,

$\sf \Rightarrow log_2 \sqrt[5]{2 \times \sqrt[3]{2\sqrt{2} } }$

$\sf \Rightarrow log_2 \sqrt[5]{2 \times \sqrt[3]{2 \times 2^{ 1 / 2 }} }$

Using law of exponents a^m × a^n= a^( m + n )

$\sf \Rightarrow log_2 \sqrt[5]{2 \times \sqrt[3]{2^{1 + ( 1 / 2) }} }$

$\sf \Rightarrow log_2 \sqrt[5]{2 \times \sqrt[3]{ 2^{ 3 / 2 }} }$

$\sf \Rightarrow log_2 \ \sqrt[5]{2 \times \bigg( 2^{ 3 / 2 } \bigg)^{1/3}} }$

Since ( a^m )^n = a^( mn )

$\sf \Rightarrow log_2 \ \sqrt[5]{2 \times \bigg( 2\bigg)^{(3/2) \times ( 1/3)}} }$

$\sf \Rightarrow log_2 \ \sqrt[5]{2 \times \bigg( 2\bigg)^{1/2}} }$

$\sf \Rightarrow log_2 \ \sqrt[5]{ \bigg( 2\bigg)^{1+( 1/2)}} }$

Using law of exponents a^m × a^n= a^( m + n )

$\sf \Rightarrow log_2 \ \sqrt[5]{ \bigg( 2\bigg)^{3/2}} }$

$\sf \Rightarrow log_2 \ \Bigg[ \bigg( 2\bigg)^{3/2}} \Bigg] ^{1/5}$

Since ( a^m )^n = a^( mn )

$\sf \Rightarrow log_2 \ \bigg( 2\bigg)^{(3/2) \times (1/5)}}$

$\sf \Rightarrow log_2 \ \bigg( 2\bigg)^{3/10}}$

Using Power rule log a^m = m.log a we get,

$\sf \Rightarrow \dfrac{3}{10} \times log_2 \ 2$

Since log_a a = 1

$\sf \Rightarrow \dfrac{3}{10}$

∴ the value of the given expression is 3/10 or 0.3.