11. \( x^{\frac{1}{6}}=2 \)

To solve the equation: \[ x^{\frac{1}{6}} = 2 \] we aim to find the value of \( x \). Here's a step-by-step solution: ### Step 1: Understand the Equation The equation involves a fractional exponent, which indicates a root. Specifically, \( x^{\frac{1}{6}} \) represents the **sixth root of \( x \)**. \[ x^{\frac{1}{6}} = \sqrt[6]{x} = 2 \] ### Step 2: Eliminate the Fractional Exponent To solve for \( x \), we want to eliminate the fractional exponent. We can do this by **raising both sides of the equation to the 6th power**. This operation will cancel out the \( \frac{1}{6} \) exponent on the left side. \[ \left( x^{\frac{1}{6}} \right)^6 = 2^6 \] ### Step 3: Simplify Both Sides When you raise a power to another power, you multiply the exponents. On the left side: \[ \left( x^{\frac{1}{6}} \right)^6 = x^{\frac{1}{6} \times 6} = x^1 = x \] On the right side: \[ 2^6 = 64 \] ### Step 4: Write the Final Solution After simplifying, the equation becomes: \[ x = 64 \] ### Verification To ensure the solution is correct, substitute \( x = 64 \) back into the original equation: \[ x^{\frac{1}{6}} = 64^{\frac{1}{6}} \] We know that: \[ 64 = 2^6 \implies 64^{\frac{1}{6}} = \left(2^6\right)^{\frac{1}{6}} = 2^{6 \times \frac{1}{6}} = 2^1 = 2 \] Which matches the right side of the original equation. ### **Answer** The solution to the equation \( x^{\frac{1}{6}} = 2 \) is: \[ x = 64 \]