Question

If $\bf {2}^{x} = {4}^{y} = {8}^{z}$
and $\bf \dfrac{1}{2x} + \dfrac{1}{4y} + \dfrac{1}{4z} = 4$ , find the

value of x.
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No Spmming ​

Answer 1

$\Large\bold{\underline{\underline{Answer:-}}}$

$\Large\bold{\boxed{\frac{7}{16}}}$

$\rule{200}{2}$

$\bf\small\bold{\underline{\underline{Step-By-Step\:Explanation:-}}}$

$\tt {2}^{x} = {4}^{y} = {8}^{z}$

Equating all the bases

4 →→2²

8 →→2³

$\tt {2}^{x} = {({2}^{2})}^{y} = {({2}^{3})}^{z}$

$\tt {2}^{x} = {2}^{2y}= {2}^{3z}$

✭By Law of exponents in above$\bf (a^m)^n = a^{mn}$ ✭

⇒ x = 2y = 3z

✭ Now bases are equal, Therefore Exponents must be equal ✭

⇒ x = 2y and x = 3z

✭We need to find values of y and z in terms of x✭

⇒ x = 2y

⇒ x/2 = y

⇒ y = x/2

So y = x/2

⇒ x = 3z

⇒ x/3 = z

⇒ z = x/3

So, z = x/3

$\tt \dfrac{1}{2x} + \dfrac{1}{4y} + \dfrac{1}{4z} = 4$

✭Substituting the value of y and z in above equation✭

$\large \displaystyle \frac{1}{2x} + \frac{1}{4( \frac{x}{2})} + \frac{1}{4( \frac{x}{3})} = 4$

$\large \displaystyle \frac{1}{2x} + \frac{1}{2(x)} + \frac{1}{\frac{4x}{3}} = 4$

$\large \displaystyle \frac{1}{2x} + \frac{1}{2x} + \frac{1}{\frac{4x}{3}} = 4$

$\large \displaystyle \frac{1}{2x} + \frac{1}{2x} +1( \frac{3}{4x}) = 4$

$\large \displaystyle \frac{1}{2x} + \frac{1}{2x} +\frac{3}{4x} = 4$

$\large \displaystyle \frac{2}{2x} +\frac{3}{4x} = 4$

Taking LCM in LHS

$\large \displaystyle \frac{2(2)}{2x(2)} +\frac{3}{4x} = 4$

$\large \displaystyle \frac{4}{4x} +\frac{3}{4x} = 4$

$\large \displaystyle \frac{7}{4x} = 4$

$\large \displaystyle \frac{4x}{7} = \frac{1}{4}$

[Reciprocal on both sides]

$\large \displaystyle 4x = \frac{1}{4} \times 7$

$\large \displaystyle 4x = \frac{7}{4}$

$\large \displaystyle x = \frac{7}{4} \div 4$

$\large \displaystyle x = \frac{7}{4} \times \frac{1}{4}$

$\large \displaystyle x = \frac{7}{16}$

Hence, value of x is 7/16

Answer 2

Step-by-step explanation:

Given that

2^x=4^y=8^z

2^x=(2^2)^y=(2^3)^z

we know (a^m)^n=a^mn so,

2^x=2^2y=2^3z

all sides bases same so cancel

we get x=2y=3z

x=2y

• x/2=y

x=3z

• x/3=z

now here

1/2x+1/4y+1/4z=4........(R)

to find X value

put y, z values in equation (R)

1/2x+1/4x/2+1/4x/3=4

1/2x+2/4x+3/4x=4

(2+2+3)/4x=4

7/4x=4

7=16x

x=7/16

therefore the value of x is 7/16.