Solve the equation (linear equation) 4^(x-7) × 8^(2x-3) =(32)/(2^(x-9) )

Question

asked Oct 10, 2020 71.7k views

2 Answers

Lebobbi · Answer 1 · 2020-10-12T10:56:02+0000

Answer:

x=4(1)/(9)

Explanation:

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

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

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

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

2^(2x-14) * 2^(6x-9) = 2^(5)(2^(-x+9))

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

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

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

9x=37

x=4(1)/(9)

Tabarek Ghassan · Answer 2 · 2020-10-14T11:42:30+0000

Answer:
x=(37)/(9)

Explanation:

By the negative exponent rule, you have that:

((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))=(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=(37)/(9)