Question

If 6^x * 8^y = 27 find the the value of x + y.​

Answer 1

Answer:

Let us simplify the numbers in powers of 2 and 3.

$\implies 6^x = 2^x \times 3^x\\\\\implies 8^y = (2^3)^y = 2^{3y}\\\\\implies 27 = 3^3$

According to the question,

$\implies 6^x \times 8^y = 27\\\\\text{Transposing }8^y \text{ to the RHS, we get:}\\\\\implies 6^x = \dfrac{27}{8^y}\\\\\implies 6^x = 27 \times 8^{-y}$

Converting the numbers into simple terms we get:

$\implies 2^x \times 3^x = 3^3 \times 2^{-3y}$

Comparing the corresponding powers of bases 2 and 3 from LHS and RHS we get:

⇒ x = -3y and x = 3

Since we know the value of x from the second equation (x=3), we get:

⇒ y = x/(-3)

⇒ y = 3/(-3)

⇒ y = -1

Hence the value of 'x' is 3 and 'y' is -1.

Therefore the value of x + y = 3 - 1 = 2.

Answer 2

$\purple {\huge \bf \underline{\underline{Question : }}}$

If 6^x * 8^y = 27 Find the the value of x + y.

$\purple {\huge \bf \underline{\underline{ Answer: }}}$

$\bf x + y = 2$

$\purple {\huge \bf \underline{\underline{ Explanation: }}}$

$\bf {6}^{ \: x} \times {8}^{ \: y} = 27 \: \: - - - - (1)$

$\bf {6}^{ \: x} = {(2 \times 3)}^{ \: x} = {2}^{ \: x} \times {3}^{ \: x}$

$\bf {8}^{ \: y} = {(2 \times 2 \times 2)}^{ \: y} = { ({2}^{3} )}^{ \: y}$

$\bf27 = {3}^{3}$

$\bf Putting \:above \: value \: in \: eq.(1)$

$\bf {2}^{ \: x} \times {3}^{ \: x} \times { ({2}^{3}) }^{ \: y} = {3}^{3} \\ \\ \bf {2}^{ \: x} \times {3}^{ \: x} = \frac { {3}^{3} }{{ {2}}^{ \: 3y}} \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \bf {2}^{ \: x} \times {3}^{ \: x} = {3}^{3} \times {2}^{ - 3y}$

Comparing the corresponding powers of bases 2 and 3 from LHS and RHS , we get :

$\boxed{\bf x = - 3y} \: \: , \: \: \boxed{ \bf x = 3}$

$\bf Putting \: x =3 \: in \:eq. \: x = - 3y$

$\bf \: 3 = - 3y \\ \\ \bf y = \frac{3}{ - 3} \\ \\ \bf y = - 1$

$\bf\therefore \: \boxed{ \bf x = 3} \: \: ,\: \: \boxed{ \bf y = - 1}$

$\bf Value \: of \: x + y \\ \\ \bf \implies3 + ( - 1) \: \: \\ \\ \bf \implies3 - 1 \: \: \: \: \: \: \: \: \\ \\ \bf \implies2 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$