求解六层嵌套根式方程

Problem

Solve:

$$\sqrt{x\sqrt{x\sqrt{x\sqrt{x\sqrt{x\sqrt{x}}}}}}=729$$

Step 1: Name the Repeated Layers

A nested square root is easier to track if each layer has a name.

Let the first layer be

$$R_1=\sqrt{x}$$

and let each new layer be

$$R_n=\sqrt{xR_{n-1}}.$$

Step 2: Rewrite Early Layers as Powers

Now rewrite the first few layers using rational exponents.

The first layer is

$$R_1=\sqrt{x}=x^{1/2}.$$

The second layer is

$$R_2=\sqrt{x\sqrt{x}}=x^{3/4}.$$

The third layer is

$$R_3=\sqrt{xR_2}=x^{7/8}.$$

Step 3: Identify the Exponent Pattern

The exponents are

$$\frac12,\frac34,\frac78,\dots$$

After $n$ layers, the exponent is

$$1-\frac{1}{2^n}.$$

So

$$R_n=x^{1-\frac{1}{2^n}}.$$

Step 4: Apply the Six-Layer Exponent

The original expression has $6$ layers, so its exponent is

$$1-\frac{1}{2^6} = 1-\frac{1}{64} = \frac{63}{64}.$$

Therefore,

$$\sqrt{x\sqrt{x\sqrt{x\sqrt{x\sqrt{x\sqrt{x}}}}}} = x^{63/64}.$$

Step 5: Set Up the Equation

Using the given equation,

$$x^{63/64}=729.$$

Step 6: Undo the Rational Power

Raise both sides to the reciprocal power $\frac{64}{63}$:

$$\left(x^{63/64}\right)^{64/63}=729^{64/63}.$$

Thus,

$$x=729^{64/63}.$$

Step 7: Express the Solution Cleanly

Since

$$729=3^6,$$

we can rewrite the answer:

$$x=(3^6)^{64/63}.$$

Using exponent rules,

$$x=3^{384/63}=3^{128/21}.$$

Therefore, the solution is

$$\boxed{x=3^{128/21}}.$$