The equation (x^x)^x = 2 ( x*x*x is equal to 2 ) is satisfied when x is equal to

by Yogi P - December 30, 2024

What is the solution to the equation (x^x)^x = 2?

A step-by-step explanation of solving the equation (𝑥^𝑥)^𝑥 = 2

Step 1: Understand the equation

The equation (𝑥^𝑥)^𝑥 = 2 can be rewritten using properties of exponents:

(x^x)^x = x^x⋅x  = x^{x^2}

Thus, the equation simplifies to:

x^{x^2} = 2

Step 2: Solve numerically

This equation involves

x raised to the power of 𝑥² , which cannot be solved algebraically. We use a numerical method to find the value of x that satisfies the equation.

Step 3: Define the function

We define the function based on the equation:

f(x) = x^{x^2} − 2

We aim to find a value of x such that

f(x) = 0

Step 4 : Why can’t we solve this like normal equations?

For equations like x² = 4, we know 𝑥 = 2 or −2. But here, x is raised to a more complicated power (𝑥²), so we can’t solve it using basic algebra. Instead, we need to use numerical methods, which involve guessing and checking values of x until we find one that works.

Step 5: Guess and check

Let’s try some values for x and see what happens:

If x = 1:

x^{x^2} = 1 ^{1^2} = 1.

That’s too small. We need 2.

If x = 2:

x^{x^2} = 2^{2^2} = 2⁴ = 16

That’s way too big.

So the solution is somewhere between 1 and 2. Let’s try a number closer to the middle, like 𝑥 = 1.5.

If x = 1.5:

x^{x^2} = 1.5 ^{1.5 ^ 2} = 1.5^2.25

This is close, but still not exactly 2. (You’d need a calculator to check this value more precisely.)

Step 4: The correct value of x Using tools like a calculator or computer (which can handle this kind of guessing and checking quickly), we find that:

x ≈ 1.414.

This number is very close to 2 (the square root of 2).

Step 5: Solution

After running the numerical solver, we get:

x ≈ 1.414.

This is approximately the value of 2.

Verification

Substitute x = √2 back into the original equation:

(√2^√2)^√2 = (√2)² = 2.

The equation holds true, confirming that x = √2 is the solution.

Conclusion

The solution to (x^x)^x = 2 is 𝑥 = √2 ≈ 1.414.