(x-27)^3/2=64

3 min read Jun 17, 2024
(x-27)^3/2=64

Solving the Equation (x-27)^(3/2) = 64

This article will guide you through solving the equation (x-27)^(3/2) = 64. We'll break down the steps and explain the reasoning behind each one.

Understanding Fractional Exponents

Before we start, let's understand what the fractional exponent (3/2) means:

  • The numerator (3) represents the power we need to raise the base to. In this case, we're cubing the base.
  • The denominator (2) represents the root we need to take. In this case, we're taking the square root.

So, (x-27)^(3/2) means we first cube (x-27) and then take the square root of the result.

Solving the Equation

  1. Isolate the base: To get rid of the fractional exponent, we first need to isolate the base (x-27). We can do this by raising both sides of the equation to the power of (2/3), which is the reciprocal of (3/2).

    [(x-27)^(3/2)]^(2/3) = 64^(2/3)
    
  2. Simplify: The exponents cancel out on the left side, leaving us with (x-27). On the right side, we need to calculate 64^(2/3). This means taking the cube root of 64 (which is 4) and then squaring the result (which is 16).

    x-27 = 16
    
  3. Solve for x: Finally, we can solve for x by adding 27 to both sides of the equation:

    x = 16 + 27
    x = 43
    

Conclusion

Therefore, the solution to the equation (x-27)^(3/2) = 64 is x = 43.

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