Question

Find the value value of $m$, if $m={64}^{-\frac{1}{2}}-(-32{)}^{-\frac{4}{5}}$.

• A. $\frac{1}{16}$
  
• B. $\frac{1}{8}$
  
• C. $\frac{3}{16}$
  
• D. $\frac{3}{8}$
  

Answer 1

The correct option is A $\frac{1}{16}$


Given, $m={64}^{-\frac{1}{2}}-(-32{)}^{-\frac{4}{5}}$.
We can write $64$ as $2^6$ and $-32$ as $(-2{)}^5$.
$\Rightarrow m={\left(2^6\right)}^{-\frac{1}{2}}-\left(\right(-2{)}^5{)}^{-\frac{4}{5}}$
And we know that $(a^m{)}^n=a^{mn}$.
$\Rightarrow m=2^{6\times (-\frac{1}{2})}-(-2{)}^{5\times (-\frac{4}{5})}$
$\Rightarrow m=2^{-3}-(-2{)}^{-4}$
Now, by using $a^{-m}=\frac{1}{a^m}$, we can write given equation as:
$\Rightarrow m=\frac{1}{2^3}-\frac{1}{(-2{)}^4}$
$\Rightarrow m=\frac{1}{8}-\frac{1}{16}[2^3=8and(-2{)}^4=16]$
$\Rightarrow m=\frac{2-1}{16}$
$\Rightarrow m=\frac{1}{16}$

Hence, the value of $m$ is $\frac{1}{16}$.