To solve the equation [tex]\((2x - 3)^3 = \frac{1}{27}\)[/tex], follow these steps:
### Step 1: Take the Cube Root of Both Sides
The first step to isolate [tex]\(2x - 3\)[/tex] is to take the cube root of both sides of the equation:
[tex]\[ (2x - 3)^3 = \frac{1}{27} \][/tex]
Taking the cube root of both sides, we get:
[tex]\[ 2x - 3 = \sqrt[3]{\frac{1}{27}} \][/tex]
Since [tex]\(\sqrt[3]{\frac{1}{27}} = \frac{1}{3}\)[/tex] (because [tex]\(\left(\frac{1}{3}\right)^3 = \frac{1}{27}\)[/tex]), the equation becomes:
[tex]\[ 2x - 3 = \frac{1}{3} \][/tex]
### Step 2: Solve for [tex]\(x\)[/tex]
Now, isolate [tex]\(x\)[/tex] by first adding 3 to both sides of the equation:
[tex]\[ 2x - 3 + 3 = \frac{1}{3} + 3 \][/tex]
Simplifying the right-hand side:
[tex]\[ 2x = \frac{1}{3} + 3 \][/tex]
Convert 3 to a fraction with the same denominator:
[tex]\[ 2x = \frac{1}{3} + \frac{9}{3} \][/tex]
[tex]\[ 2x = \frac{1 + 9}{3} \][/tex]
[tex]\[ 2x = \frac{10}{3} \][/tex]
Now, divide both sides by 2:
[tex]\[ x = \frac{10}{3} / 2 \][/tex]
[tex]\[ x = \frac{10}{3} \cdot \frac{1}{2} \][/tex]
[tex]\[ x = \frac{10}{6} \][/tex]
[tex]\[ x = \frac{5}{3} \][/tex]
So, one real solution is:
[tex]\[ x = \frac{5}{3}, \text{ which is approximately } 1.66666666666667 \][/tex]
### Step 3: Consider the Complex Solutions
The equation [tex]\((2x - 3)^3 = \frac{1}{27}\)[/tex] is a cubic equation. A cubic equation generally has three roots—which may include real and complex roots.
Besides the real root [tex]\( x = 1.66666666666667 \)[/tex], there are also two complex roots. These complex roots are:
1. [tex]\( x = 1.41666666666667 - 0.144337567297406i \)[/tex]
2. [tex]\( x = 1.41666666666667 + 0.144337567297406i \)[/tex]
### Summary of Solutions
The three solutions to the equation [tex]\((2x - 3)^3 = \frac{1}{27}\)[/tex] are:
1. [tex]\( x = 1.66666666666667 \)[/tex]
2. [tex]\( x = 1.41666666666667 - 0.144337567297406i \)[/tex]
3. [tex]\( x = 1.41666666666667 + 0.144337567297406i \)[/tex]
