Solve the equation with rational exponents. (X-1)^2/3=36
raise both sides to the 3/2 power:
((x-1)^2/3)^3/2 = x-1
So, x-1 = 36^3/2 = 6^3 = 216
x = 217
4rrr
To solve the equation with rational exponents, we need to isolate the base with the rational exponent and eliminate the exponent by raising both sides of the equation to the reciprocal of the exponent.
In this case, we have the equation: \((x-1)^{\frac{2}{3}} = 36\).
To eliminate the exponent, we can raise both sides of the equation to the reciprocal of the exponent, which is \(\frac{3}{2}\). This results in:
\[\left((x-1)^{\frac{2}{3}}\right)^{\frac{3}{2}} = 36^{\frac{3}{2}}\].
Applying the exponent rule \((a^m)^n = a^{mn}\), we simplify the equation to:
\[(x-1)^1 = 36^{\frac{3}{2}}\].
Simplifying further, we have:
\[(x-1) = (36^{\frac{3}{2}})^1\].
Next, we evaluate the exponent on the right-hand side of the equation:
\[(x-1) = (6^2)^3\].
Squaring 6 gives us:
\[(x-1) = 6^6\].
Finally, we have the equation:
\[x-1 = 46656\].
To solve for x, we add 1 to both sides of the equation:
\[x = 46656 + 1\].
Thus, the solution to the equation is:
\[x = 46657\].
Therefore, the value of x that satisfies the equation \((x-1)^{\frac{2}{3}} = 36\) is 46657.