Question

Solve the equation (linear equation)

Answer 1

Answer: $x=\frac{37}{9}$

Step-by-step explanation:

By the negative exponent rule, you have that:

$(\frac{1}{a})^n=a^{-n}$

By the exponents properties, you know that:

$(m^n)^l=m^{(nl)}$

$(m^n)(m^l)=m^{(n+l)}$

Rewrite 4, 8 and 32 as following:

4=2²

8=2³

32=2⁵

Rewrite the expression:

$(2^2)^{(x-7)}*(2^3)^{(2x-3)}=\frac{32}{2^{(x-9)}}$

Keeping on mind the exponents properties, you have:

$(2)^{2(x-7)}*(2)^{3(2x-3)}=32(2^{-(x-9)}$

$(2)^{2(x-7)}*(2)^{3(2x-3)}=(2^5)(2^{-(x-9)})\\\\(2)^{(2x-14)}*(2)^{(6x-9)}=(2^5)(2^{(-x+9)})\\\\2^{((2x-14)+(6x-9))}=2^{(5+(-x+9))}$

As the bases are equal, then:

$(2x-14)+(6x-9)=5+(-x+9)\\\\2x-14+6x-9=5-x+9\\\\8x-23=14-x\\9x=37$

$x=\frac{37}{9}$

Answer 2

Answer:

$x=4\frac{1}{9}$

Step-by-step explanation:

We are given the following linear equation and we are to solve it:

$4 ^ { x - 7 } \times 8 ^ { 2x - 3 } = \frac { 32 } { x ^ { x - 8 } }$

Changing the constants to the same base to make it easier to solve:

$2 ^ { 2 ( x - 7 ) } \times 2^ { 3 ( 2x - 3 ) } = \frac { 2 ^ 5 } { 2 ^ { x - 9 } }$

$2^{2x-14} \times 2^{6x-9} = 2^{5}(2^{-x+9})$

$2 ^ { 2x - 14 + 6x - 9 } = 2 ^ { 5 - x + 9 }$

$2 x + 6x - 14 - 9 = 5 - x + 9$

$8x+x=14+9+5+9$

$9x=37$

$x=4\frac{1}{9}$