Interesting Exponents

This can be written as:

$2^{3m} = 27$

Therefore:

${ ({ 2 }^{ 2m }) }^{ \frac { 3 }{ 2 } } = 27$

$\therefore 2^{2m} = 4^{m} = 27^{\frac{2}{3}} = 3^{2} = \boxed{9}$

$8^m=27$
$\Rightarrow(2^3)^m=27$
$\Rightarrow(2^m)^3=3^3$
$\Rightarrow 2^m=3$
$\Rightarrow(2^m)^2=3^2$ (Squaring both sides)
$\Rightarrow(2^2)^m=9$
$\Rightarrow 4^m=9$

$8^{m} = (4 \times 2)^{m} = 4^{m} \times 2^{m}$

$4^{m} \times 2^{m} = 27$

So, $4^{m} = \frac{27}{2^{m}}$ .

If: $m = log_8 27$

then: $4^{m} = \frac{27}{2^{log_8 27}} = \frac{27}{3} = \boxed{9}$ .

Alternatively: $4^{m} = 4^{log_8 27} = \boxed{9}$

$8^{m}=27$ ----> $2^{3m} = 3^3$

$3m.ln 2 = 3.ln 3$

$m.ln 2 = ln 3$

$2^m = 3$

$\boxed{4^m = 2^m.2^m = 3x3 = 9}$

2^3m=3^3 2^2m=3^2 so, 9

To find the value of $m$ , we must take the logarithm in base 8 of each side and work from there onwards:

$\log _{ 8 }{ { 8 }^{ m } = \log _{ 8 }{ 27 } }$

Due to the Power Property of logarithms ( $\log _{ b }{ { m }^{ n } } =n(\log _{ b }{ m } )$ ) we are able to simplify the left side of the equation:

$\log _{ 8 }{ { 8 }^{ m } } =m(\log _{ 8 }{ 8 } )=m(1)=m$

For now, we will express the right side as simply $log _{ 8 }{ 27 }$ .

After simplifying this equation, we know that $m= \log_{8}{27}$ .

The question asks what the value of ${ 4 }^{ m }$ is. To find it, we have to plug our value of $m$ back into this expression:

${ 4 }^{ (\log _{ 8 }{ 27 } ) }$ of which the answer is $\boxed{9}$

$8^m = 27 \\\\ \log_8 27 = m \\\\ \log_{2^3} 3^3 = m \\\\ \frac{1}{3} \cdot 3 \cdot \log_2 3 = m \\\\ \log_2 3 = m \\\\ 2^m = 3 \\\\ (2^m)^2 = (3)^2 \\\\ 2^{2m} = 9 \\\\ (2^2)^m = 9 \\\\ \boxed{4^m = 9}$

Given\quad that\\ \qquad \qquad { 8 }^{ m }={ 27 }\\ Step\quad I\\ \qquad Take\quad cube\quad root\quad on\quad both\quad sides.\\ \qquad \quad \quad { { (8 }^{ m }) }^{ \frac { 1 }{ 3 } }={ 27 }^{ \frac { 1 }{ 3 } }\quad =>\quad { { (2 }^{ 3m }) }^{ \frac { 1 }{ 3 } }={ ({ 3 }^{ 3 }) }^{ \frac { 1 }{ 3 } }\\ or\qquad \boxed { { 2 }^{ m }={ 3 } } \\ Step\quad II\\ \qquad Now\quad take\quad squre\quad on\quad both\quad sides\\ \qquad { { ( }2^{ m }) }^{ 2 }={ 3 }^{ 2 }\\ we\quad get\quad \boxed { { 4 }^{ m }=9 } \\

${ 8 }^{ m }={ ({ 2 }^{ 3 }) }^{ m }={ ({ 2 }^{ m }) }^{ 3 }=27\quad \Rightarrow \quad { 2 }^{ m }=3\quad \Rightarrow \quad { ({ 2 }^{ m }) }^{ 2 }={ ({ 2 }^{ 2 }) }^{ m }={ 4 }^{ m }=9$

A fun one!