(x-1)^3/4=64

Jasonbradley

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Jun 17, 2024

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Solving the Equation: (x-1)^(3/4) = 64

This article will guide you through solving the equation (x-1)^(3/4) = 64.

Understanding Fractional Exponents

Before we begin, let's understand what a fractional exponent means. In general, x^(m/n) is equivalent to the n-th root of x raised to the power of m. In our case, (x-1)^(3/4) represents the fourth root of (x-1) cubed.

Solving the Equation

1. Isolate the base: To get rid of the fractional exponent, we need to raise both sides of the equation to the power of 4/3. This is the reciprocal of 3/4.

   ( (x-1)^(3/4) )^(4/3) = 64^(4/3)

   This simplifies to: x-1 = 64^(4/3)

2. Simplify the right side: Calculate 64^(4/3). This is equivalent to the cube root of 64 raised to the power of 4.

   64^(4/3) = (∛64)^4 = 4^4 = 256

3. Solve for x: Now the equation is: x - 1 = 256. Adding 1 to both sides gives us: x = 257.

Conclusion

Therefore, the solution to the equation (x-1)^(3/4) = 64 is x = 257.