Question

a^2m-5*b^3n-2*c^4p-1=abc where a b c are rational numbers , find the value of 1/mnp

Answer 1

Given :  $a^{2m-1}.b^{3n-2}.c^{4p-1} = abc$   , a^2m-5*b^3n-2*c^4p-1=abc  

To find :  value of 1/mnp

Solution:

a^2m-5*b^3n-2*c^4p-1=abc

$a^{2m-1}.b^{3n-2}.c^{4p-1} = abc$

abc = a¹b¹c¹

assuming a, b & c are coprime

Hence Equating power of a with a , b with b c with c

2m - 5 = 1  => 2m = 6 => m = 3

3n - 2 = 1 => 3n = 3 =>  n = 1

4p - 1 = 1 => 4p = 2 => p = 1/2

mnp = (3)(1)(1/2)  =  3/2

1/mnp  = 1/(3/2) = 2/3

1/mnp = 2/3

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Answer 2

Given:

$\bold{a^{2m-5}b^{3n-2}c^{4p-1}=abc}$

Find:

The value of=  $\bold{\frac{1}{mnp}}$

Solution:

value:

$\bold{a^{2m-5}b^{3n-2}c^{4p-1}=abc}$

if abc = $a^1 \times b^1\times c^1$

if the above value is coprime then:  

$\Rightarrow \bold{2m-5=1}\\\Rightarrow 2m=1+5\\\Rightarrow 2m= 6\\\Rightarrow m= \frac{6}{2}\\\Rightarrow m= 3$

$\Rightarrow \bold{3n-2=1}\\\Rightarrow 3n=1+2\\\Rightarrow 3n=3\\\Rightarrow n=\frac{3}{3}\\\Rightarrow n= 1$

$\Rightarrow \bold{4p-1= 1}\\\Rightarrow 4p= 1+1\\\Rightarrow p= \frac{2}{4}\\\Rightarrow p= \frac{1}{2}$

Calculating the value of:  $\bold{\frac{1}{mnp}}$

$\Rightarrow \bold{\frac{1}{mnp}} = \frac{1}{ 3\times 1\times \frac{1}{2}}$

            $= \frac{1}{ \frac{3}{2}}\\\\= \frac{2}{3}$

The final answer is "2/3".