Questions: Solve the equation with rational exponents. (x-1)^(2 / 3)=36

Solution Steps

To solve the equation \((x-1)^{2/3} = 36\), we need to eliminate the rational exponent by raising both sides of the equation to the reciprocal of \(\frac{2}{3}\), which is \(\frac{3}{2}\). This will isolate \(x-1\) on one side of the equation. After that, we can solve for \(x\) by adding 1 to both sides.

To solve the equation \((x-1)^{2/3} = 36\), we first eliminate the rational exponent by raising both sides to the power of \(\frac{3}{2}\). This gives us: \[ (x-1) = 36^{\frac{3}{2}} \]

Calculate \(36^{\frac{3}{2}}\): \[ 36^{\frac{3}{2}} = (36^{\frac{1}{2}})^3 = 6^3 = 216 \] Thus, we have: \[ x-1 = 216 \]

Add 1 to both sides to solve for \(x\): \[ x = 216 + 1 = 217 \]

Final Answer