Question

Chapter - Laws of Indices

If $(222.2)^{a}=(44.44)^{b}=(8.888)^{c}$, then show that $\frac{1}{a}+\frac{1}{c}=\frac{2}{b}$.​

Answer 1

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♦ Given that

>> $(222.2)^a = (44.44)^b = (8.888)^c$

♦ Then to show

$\dfrac{1}{a} + \dfrac{1}{c} = \dfrac{2}{b}$

♦ Let us consider the Numbers

• We can write

>> $(44.44)^b \:as \: ((222.2)^2)^b = (222.2)^{2b}$

>> $(8.888)^c \: as\: ((222.2)^3)^c = (222.2)^{3c}$

♦ Now as

• $(222.2)^a = (44.44)^b = (8.888)^c$

$\longrightarrow (222.2)^a = (222.2)^{2b} = (222.2)^{3c}$

>> By comparing bases we can say that .

$\longrightarrow a = 2b = 3c$

$\implies c = \dfrac{a}{3}$

$\implies b = \dfrac{a}{2}$

♦ Then by substituting the aquired value of c in

$\dfrac{1}{a} + \dfrac{1}{c} = \dfrac{2}{b}$

$\implies \dfrac{1}{a} + \dfrac{1}{\dfrac{a}{3}} = \dfrac{2}{b}$

$\implies\dfrac{1}{a} + (\dfrac{1}{1}\times \dfrac{3}{a}) = \dfrac{2}{b}$

$\implies \dfrac{1}{a} + \dfrac{3}{a} =\dfrac{2}{b}$

$\implies \dfrac{4}{a} = \dfrac{2}{b}$

$\implies \dfrac{2}{\dfrac{a}{2}} = \dfrac{2}{b}$

♦As b = $\dfrac{a}{2}$

$\implies \dfrac{2}{b} = \dfrac{2}{b}$

♦ As RHS = LHS , Hence Shown .

Answer 2

Answer:

1 / a + 1 / c = 2 / b  

Step-by-step explanation:

Given,  

     ( 222.2 )^a = ( 44.44 )^b = ( 8.888 )^c

Let,

     ( 222.2 )^a = ( 44.44 )^b = ( 8.888 )^2 = k  

Thus,

⇒ ( 222.2 )^a = k  

⇒ 222.2 = k^( 1 / a )            …( i )

⇒ ( 44.44 )^b = k  

⇒ 44.44 = k^( 1 / b )              …( ii )

⇒ ( 8.888 )^c = k  

⇒ 8.888 = k^( 1 / c )              …( iii )

Multiply ( i )  and  ( iii ) : -  

= > k^( 1 / a ) x k^( 1 / c ) = 222.2 x 8.888

= > k^( 1 / a + 1 / c ) = 2 x 111.1 x 8 x 1.111

= > k^( 1 / a + 1 / c ) = 2 x 2^3 x 100 x ( 1.111 )^2

= > k^( 1 / a + 1 / c ) = 2^4 + 10^2 x ( 1.111 )^2

= > k^( 1 / a + 1 / c ) = { 2^2 x 10 x 1.111 }^2  

= > k^( 1 / a + 1 / c ) = ( 44.44 )^2

= > k^( 1 / a + 1 / c ) = { k^( 1 / b ) }^2                  { from ( ii ) }

= > k^( 1 / a + 1 / c ) = k^( 2 / b )  

From the properties of indices, we know, if bases are same, powers are equal.  

Then,  

= > 1 / a + 1 / c = 2 / b  

Hence proved.