Question

Evaluate


`{3}^(x) - {3}^(x - 2) = 24`
Evalutae this formula and solve for x​

Answer 1

`x = 3`

Explanation:

Given equation is ,

3^x - 3^{x-2} = 24

we can write it as,

3^x - (3^x/3^2) = 24

take out 3^x as common,

3^x ( 1 - 1/3^2) = 24

simplify,

3^x (1 -1/9)=24

3^x (9-1/9)=24

3^x * 8/9 = 24

3^x = 24 * 9/8

3^x = 27

3^x = 3^3

on comparing,

x = 3

and we are done!

Answer 2

x = 3

Explanation:

Given equation:

`3^x-3^(x-2)=24`

Rewrite the exponent of the first term as (x - 2 + 2):

`3^(x-2+2)-3^(x-2)=24`

`\textsf{Apply the exponent rule} \quad a^(b+c)=a^b \cdot a^c:`

`3^((x-2)+2)-3^(x-2)=24`

`3^(x-2)\cdot 3^2-3^(x-2)=24`

`\textsf{Factor out the common term $3^(x-2)$}:`

`3^(x-2)(3^2-1)=24`

Simplify the brackets:

`3^(x-2)\cdot 8=24`

Divide both sides by 8:

`3^(x-2)=3`

Apply the exponent rule a = a¹ :

`3^(x-2)=3^1`

`\textsf{Apply the exponent rule} \quad a^(f(x))=a^(g(x)) \implies f(x)=g(x):`

`x-2=1`

Add 2 to both sides of the equation:

`x=3`