Certainly! Let's solve the equation step by step:
Given:
[tex]\[ \frac{1}{\sqrt[3]{2x - 3}} = 2 \][/tex]
Step 1: Get rid of the fraction by taking the reciprocal on both sides of the equation.
[tex]\[ \sqrt[3]{2x - 3} = \frac{1}{2} \][/tex]
Step 2: Cube both sides to remove the cube root.
[tex]\[ (\sqrt[3]{2x - 3})^3 = \left( \frac{1}{2} \right)^3 \][/tex]
[tex]\[ 2x - 3 = \frac{1}{8} \][/tex]
Step 3: Isolate [tex]\( x \)[/tex] by solving the resulting linear equation:
[tex]\[ 2x - 3 = \frac{1}{8} \][/tex]
Step 4: Add 3 to both sides to move the constant term to the right-hand side:
[tex]\[ 2x = \frac{1}{8} + 3 \][/tex]
Step 5: Convert 3 to a fraction with the same denominator as [tex]\(\frac{1}{8}\)[/tex]:
[tex]\[ 3 = \frac{24}{8} \][/tex]
[tex]\[ 2x = \frac{1}{8} + \frac{24}{8} \][/tex]
Step 6: Combine the fractions:
[tex]\[ 2x = \frac{25}{8} \][/tex]
Step 7: Divide both sides by 2 to isolate [tex]\( x \)[/tex]:
[tex]\[ x = \frac{25}{8} \times \frac{1}{2} \][/tex]
[tex]\[ x = \frac{25}{16} \][/tex]
Step 8: Convert the fraction to a decimal for a more intuitive answer:
[tex]\[ x = 1.5625 \][/tex]
Thus, the solution to the equation [tex]\(\frac{1}{\sqrt[3]{2 x-3}}=2\)[/tex] is:
[tex]\[ x = 1.5625 \][/tex]
This completes the step-by-step solution of the given equation.
